as long as we paid the semester fee the university never bat an eye

Public universities usually give you a lot less time, think months, and one may also be constrained by visa allowances. Not sure if it's also the case in the US.

Not sure how is it with PhD, but with Masters and below, my experience (context: public universities, Poland) is that even in a public university you can get quite a bit of extra time if you... just ask for it. Preferably not in the last minute. The staff doesn't have any particularly strong reason to kick you out just for failing to meet a deadline, or even failing a class - the rules can be bent quite a lot if they see you're making an effort and want to complete your thesis / learn something.

Just as long as you don't hit an administrative blocker, that is. In my case, I never submitted the thesis, because my request to extend the time coincided with the faculty being split in two and completely overhauled - so instead of extension, I was told that, because of "syllabus changes", I would need to repeat the entire MSc level (2 years) to be able to submit the thesis now. I politely declined.