When I, as an American, arrive at a hotel or rental in Mexico, I must present, or at that time process a visa entry form. That and my passport are logged by the provider.
One can assume they file that form with the government.
If I try to book further stays, beyond the allowable limit, the booking would be blocked, and I would be in trouble with the government / informed I need to leave promptly.
Think more flexibly dude, other places are already handling this.
Because you are so rigid in your thinking, let’s drop the word “visa” and construct a new idea (uh oh!) And call it a “tourism allowance”