I just want to say that these are unrelated: opening a bank account doesn't actually require a proof of residence in France by law, it's the banks themselves that often require it (but when you move abroad they are generally happy to update your address without proof), needing a check for renting is absolutely infuriating but also not a legal requirement, only something private landlords usually ask (publicly subsided housing and student accommodation organisations that regularly see foreign students generally allow other means of guarantee).

So these are entirely problems with private actors, not the French administration - you will find that the French administration is often very helpful when you need to pay things to them, most means of payment are possible, foreign credit cards are no problem for example, and they even take American Express for paying fines.

And wealthfare fraud is not conducted with checks, although it might require false proofs of residence.