The idea of replacing a 'gravitational force' with spacetime curvature gave us General Relativity; extending this same idea to electromagnetism gives us Kaluza-Klein theory https://en.wikipedia.org/wiki/Kaluza%E2%80%93Klein_theory
In QFT, "particles" and "forces" are emergent phenomena (waves of excitation in the underlying fields, and the couplings/interactions/symmetries of those fields). QFT tends to be modelled using Lagrangian mechanics too.
An attempt at a true ELI5 is the bodies exist in what we know as spacetime, not as separate independent concepts of space and time which we perceive from our day to day experience, so we have to know a bit about the difference. Chiefly in spacetime everything always travels the same "speed" (c, the universal speed limit) and it's just a matter of how much of that speed appears as "traveling through space" and how much appears as "traveling through time". When 2 bodies warp spacetime it causes changes in the way each body's spacetime speed is distributed causing them to accelerate towards each other.
The ELI15 version is think about vectors in our normal concept of 3D space first, if I told you a body was always moving at 100 meters per second and it was 100% in the horizontal direction you'd say there was 0 meters per second in the vertical direction. Now say something curves this geometry a little bit, the body will still be traveling at 100 meters per second but now a tiny bit of that speed may appear to manifest in the vertical direction and a tiny bit less appear to manifest in the horizontal direction. Same general story with spacetime except the math is a lot more complex leading to some nuance in how things actually change.
The ELI20 version should you want to understand how to calculate the effects yourself is probably best left to this 8 part mini series rather than me https://youtu.be/xodtfM1r9FA and the 8th episode recap actually has a challenge problem to calculate what causes a stationary satellite to fall to the sun (in an idealized example) that exactly matches your question.
It's 1:1 with the relations in the equations up until the analogy of warped Euclidean space changing the vector at which point the description is functionally very similar but relativity follows very different (but also somewhat similar in a way) mathematical mechanics to the vector changing.
The "spacetime speed vector" is more formally the four-velocity and it's true that the norm of this 3 component space 1 component time vector is strictly tied to c. At the same time the four-velocity doesn't actually mathematically behave like a euclidean vector space vector where you can just add another like vector describing the effects of the warping and call it a day. In reality you have to run it through the metric tensor first (some function for the given instance that describes the geometry of warped spacetime) to get things in a coordinate space that is usable. Once you have that you actually have to run it through the geodesic equation to see what the acceleration will be as using the mapped four-vector alone will only tell you about the current velocity components in your coordinate space not the effect of the spacetime warping on something in them. These kinds of differences are the bits I swept under the rug as "nuance in how things actually change" but the net concept of the four-vector shifting components due to the warping of spacetime as an object moves along its world line is 100% the net result.
Also I can't really take credit for the method of explanation, just some of the simplified wording. I do find this explanation not only infinitely more accurate but actually easier to understand than the damn rubber sheet analogies or even improved/3D space warping analogies as they still leave out the time portion of the spacetime gradient which actually plays a bigger role in these examples.
It's spacetime curvature. This is an important distinction, because although you can zero out the spatial component of your 4-vector you can't also zero out the time component.
Apparently you can think of the gravitational force as arising from time gradients [1]. Time flows slower closer to the planet, so if your arm is pointing towards the planet then your arm is advancing slightly slower in a particular way and this creates a situation where your arm wants to pull away from you; an apparent force.
A common framework for explaining spacetime gravitation is the rubber sheet with a heavy ball, showing that other objects on the sheet fall towards the ball. This is really flawed because it explains gravity using gravity.
Instead, you keep the rubber sheet and the single ball. Instead of placing other objects on the curved rubber, project (using a projector if you want) a straight line (from a flat surface) down onto the rubber. If you trace the projection of the line onto the rubber, you'll notice that it is no longer straight - it curves with the rubber (especially if you subsequently flatten the rubber out). That's a world line[1]. That's the direction of movement that an object would see as its "momentum" - but it wouldn't actually follow the world line, as the world line changes when the object moves.
To build a geodesic (the actual orbit/movement of the object), you need to move along the world line and then build a new one, repeatedly. I haven't completely figured out the instructions to build a geodesic in this analogy, but seeing/imagining the curved world line should be enlightening:
Think of your velocity vector as having a time component. The magnitude of this vector is c, so when you are at rest, you're moving full speed through time. When you accelerate, you shift some of this speed into the spatial dimensions. This is also why time passes more slowly for moving objects. Gravity also has this effect because not only is space curved, but space-time is curved. This means what would normally be a straight path through time is partially warped into the spatial dimensions when you encounter such a curvature.
There is no such attraction, same asyour question doesn't make sense. The Delta v has to do with the net force, what's actually happening, but this "attraction" is described as "what if you took away one of the forces impacting this"
For a curvature based model, the delta v being 0 means that the gradients around each body are equal to each other, but that doesn't say anything about what's causing those gradients.
To find this "attraction", you have to calculate the curvature while leaving some sources out
Imagine a 2d sheet that is weiged by steel balls. It'll be curved because of weights. Now, put a sand on it and it'll start rolling according to sheet's curvature. That's attraction between bodies for you.
Essentially for any given spacetime we can calculate out geodesics for any freely-falling object; it's just the path these objects follow through spacetime unless otherwise disturbed. Here we're interested in such objects that couple only to gravitation. These "test objects" do not radiate at all, not even when brought into contact with each other, and they don't absorb radiation. They don't attract electromagnetically, or feel electromagnetic attraction, and they don't feel such repulsion either. They also don't feel the weak or strong interactions. So they're always in free-fall -- always in geodesic motion -- because they can't "land" on anything.
We take one further step into fiction and prevent these test objects from generating curvature themselves. You can fill flat spacetime with them, and spacetime will stay flat. This is completely unphysical, but it's a handy property for exploring General Relativity.
If we put such an object into flat spacetime, we can use it to define a set of spacetime-filling extended Cartesian coordinates, where we add time to the Cartesian x, y, and z labels. We set things up so that the object is always at x=0, y=0, z=0, but can be found at t < 0 and t=0 and t > 0. The units are totally arbitrary. You can use SI units of seconds and metres, or seconds and light-seconds, or microseconds and furlongs: for our purposes it doesn't matter.
We can introduce another such object offset a bit, so that it is found initially at t=0,x=200,y=0,z=0. Again, the units are unimportant, it only matters that the second object is not at the same place as the first. This object is set up to always be at y=0,z=0.
In perfectly flat spacetime, these two objects, for t=anything, will be found at x=0 and x=200 respectively, and always at y=0, z=0.
They do not converge, ever, not in the past or in the future. They also do not diverge. The choice of coordinates doesn't matter any more than the choice of units; we could change the picture to keep the second object always at x=200, and the first will move from x=0. Or we can let them both wander back and forth along x, but with constant separation. But let's stick with our first choice of holding the first particle at the spatial origin at all times.
Now, what happens if we give the first object a little bit of stress-energy (you can think of that as mass in this setup)?
The geodesics generated now are not those of flat spacetime, but rather much closer to those of Schwarzschild. We have perturbed flat spacetime with the nonzero mass.
The first object, if we keep it always at x=0,y=0,z=0 now causes the second object to be on a new geodesic that is x != 200 at different times. Depending on the relationship between the "central mass" at the origin and distance x=200, the geodesic evolution of x for all t for the second object might look like an elliptical, circular, or hyperbolic trajectory [1].
If on the other hand we give both objects the same mass, we end up calculating out geodesics that focus. There will be at least one time t > 0 where the test objects will occupy the same point in spacetime, t=?,x=?,y=0,z=0. (This is called a "caustic").
Raychauduhri showed that caustics are highly generic[2]: you need electromagnetic repulsion (which means a global charge imbalance, which is not a feature of our universe); strong gravitational radiation (which is not a feature of our universe except perhaps in the extremely early universe); or a metric expansion of space (which is a feature of our universe, and leads to large volumes in which geodesics diverge, avoiding caustics, and small volumes in which geodesics converge such that caustics are only avoided by non-gravitational interactions).
This is the General Relativistic picture of masses attracting each other: objects follow geodesics unless shoved off them (by e.g. electromagnetic interaction), or until they "land" on something; in most physically plausible spacetimes there are generically intersecting geodesics and most things find themselves on one; and so close approaches, collisions, mergers, and so forth are practically inevitable.
Lastly, consider an https://en.wikipedia.org/wiki/Accelerometer . A calibrated one in free-fall anywhere should always report "0"; dropping the same out of an airplane should show a slight upwards acceleration imparted by collisions with the air, and then a big upwards one upon contact with the surface. These collisions with air molecules and water or ground molecules shove the falling accelerometer off its geodesic. An accelerometer resting on the ground or on the airplane will show an acceleration somewhere around 10 m/s^2 in SI units: it is being pushed off free-fall by interactions.
Two accelerometers freely-falling in flat spacetime will eventually collide with one another thanks to the focusing theorem. Only as they collide will the accelerometers show nonzero.
Finally, you can even experiment with this yourself: install https://phyphox.org/ on a modern smartphone and rest it on the floor, take it with you into an elevator, jump up and down, or throw it a long way (try not to break it, and try to avoid it rotating much while in the air) and you'll see that when in flight it registers a near-zero acceleration, but a substantial acceleration when in your hand as you wind up and throw, and a substantial acceleration when it lands. While in the air your phone is in practically-geodesic-motion.
It's this property of free-fall -- the absence of acceleration, even if one is orbiting or falling straight towards some massive object -- that is at the root of Einstein's gravitation, and which distinguishes it from Newton's gravity. It is formalized into the https://en.wikipedia.org/wiki/Equivalence_principle .
Although your thrown phone and the Earth are interacting gravitationally, neither the phone nor the planet feels a "pull" towards one another during the phone's flight, or during a parachutist's drop. The geodesics generated around the freely-falling Earth and (effectively) freely-falling phone just lead to greater radial motion by the phone.
The idea of replacing a 'gravitational force' with spacetime curvature gave us General Relativity; extending this same idea to electromagnetism gives us Kaluza-Klein theory https://en.wikipedia.org/wiki/Kaluza%E2%80%93Klein_theory
The current state of the art is Quantum Field Theory (of which the Standard Model is an example) https://en.wikipedia.org/wiki/Quantum_field_theory
In QFT, "particles" and "forces" are emergent phenomena (waves of excitation in the underlying fields, and the couplings/interactions/symmetries of those fields). QFT tends to be modelled using Lagrangian mechanics too.