Ok, now prove that they are shitty so you can fire them.

That “complete nonsense” is what US employment lawyers would advise you. But hey, some French guy on the internet knows more.

That 'French' guy owned a US company at some point.

owning a company sure takes a lot of ... file a document

Attempting to circumvent anti discrimination laws by documenting and firing over technical violations only for the class you aren't allowed to discriminate against doesn't improve the legal situation vs simply discriminating more straightforwardly and firing without cause. Firing for cause rather than without can avoid severance payout though.

In theory but most plaintiffs employment lawyers (working on contingency) aren’t going to invest that much time and money for a run of the mill anti discrimination lawsuit.