You’ve already answered your own question.

But don’t forget culture. The USA was founded by the religious extremists we (the UK) booted out hundreds of years ago.

There is a very deep, biblical/puritanical streak running through the psyche of Americans (and to a lesser extent the English speaking world) that demands extreme punishment for wrongdoing. Rehabilitation is a dirty word, it doesn’t matter if someone commits a crime again, the whole point is that if they do they rot in prison.