Well, no, because speeding increases the energy of potential collisions, and also encourages others to speed. If you have to drive more than once, "speeding" isn't necessarily a dominant strategy.
When driving, you want to keep your velocity relative to other cars as low as possible, while retaining your ability to avoid obstacles / not murder cyclists and pedestrians.
If cars could go at mach 3, then you'd want to avoid roads entirely: it's not feasible to avoid obstacles at that speed, and you certainly don't want to get hit by shockwaves or – worse – actual cars travelling at such relative velocities. So no, you would have worse problems in this hypothetical.
Yes, or relativistic velocities. I figured that harm due to the energy of the collision is limited by death, while time spent on road tends to zero, but your mention of shockwaves brings to mind that a sufficiently high-energy collision would also be a danger to otherwise uninvolved road users, or residents of nearby towns, so my first assumption was incorrect. (I leave aside the problem of maintaining control of a mach 3 road vehicle as an implementation detail.)
