Usually no. If you damage the connector it tears the traces off the board irreparably. If it doesn't do that then there are two rows of pins which need to be resoldered. One row is under the connector. It is near impossible to solder those properly without non trivial equipment. Usually if you get a repair they solder only one pair of pins to the board and that means your USB connector only works plugged in one way up.

This is one reason amongst many that I'm not a particularly big fan of USB-C

Nope, and it's more and more common. I had to trash a $2k monitor due to faulty USB C connector. They require full board replacement.

What stops you from desoldering the bad connector and soldering on a new one?

SMD rework sucks if you're not used to it.

It sucks if you are used to it!

Not in my experience.

But dense SMD connectors with heat sensitive parts and damaged boards more than just suck.

Assuming you can find the part "full board replacement" may not be as bad as it sounds for a monitor. Your main point still stands though, ideally a faulty connector shouldn't require to replace anything besides the connector itself.