I'm an American, and I disagree, mostly. I've never heard someone say a "flat wheel"; it's always flat tire. Also if an American says a wheel came off when they mean a tire, they misspoke/misunderstand what they are talking about. There are plenty of Americans that know next to nothing about their vehicle despite it usually being the most expensive or second most expensive thing they own.

Disclaimer: Midwest USA here, can't speak for other parts of the US.