Just return to old fashioned styles of universities with tutorials, lectures, offline handwritten exams, and viva voce.
It's very hard to hide the fact that someone else did an assignment when you have to defend it in front your tutor and a small group of fellow students and it's next to impossible to pass a final viva without knowing and understanding what you are talking about.
The problem is we have all become addicted to cheap 'education' and a the traditional methods are expensive.
But I think the institutions and the students need to ask themselves what the university is for. Is it to hand out diplomas or is it there so that the students can learn? A student who only wants the diploma has an incentive to cheat, one who wants to learn does not because the only person cheated is themself.
I think your last point is precisely why universities shouldn't limit access to llms beyond reasonable means. Make it hard for the weak to access, and easy enough for those dedicated. the ones to make an effort to cheat aren't there to learn anyway
so we should just let them through?