Just failed my driving test

I'd estimate that I see dented and smashed up cars driving on the US roads about 5 times as often as I do in the UK.

That's probably because the US doesn't have the equivalent of MOT testing which seems to be much pickier about the driving standard of cars as opposed to the US emissions tests which, if memory serves are not as frequently required.
A lot of people seem to be able to drive any old piece of crap in the US that would never pass a UK MOT.  Just watch the show Pimp my Ride for proof!

Vehicle inspections certainly vary from state to state, with some places have no official testing at all.

I understand that Florida once introduced statewide inspections but they were so unpopular that they were abandoned.  (Can't imagine that happening here!)   So today the state just has emissions inspections in some of more urbanized counties (Dade etc.) and no tests at all elsewhere.

A vehicle with dents or smashes would pass an MOT over here unless it interfered with the operation or safety of the car - the dents and smashes I'm talking about in the US are similar, I don't think it's anything to do with no MOTs - although, I'm not sure how good insurance is over there, may be that most people's insurance doesn't cover damage to their own car?  Still a lot of dented cars!

A vehicle with dents or smashes would pass an MOT over here unless it interfered with the operation or safety of the car

True.  It would have to be something like a smashed panel fouling part of the suspension, or a jagged edge which could cause injury.