I really don't understand your reasoning. I think it was a perfectly reasonable question.
The all-sets-finite system is a possible axiom system, and I really think that it must have been studied. I remember hearing about it somewhere, but I just can't remember, and couldn't find in Google.
It is very clearly NOT a "too localized" question.
It is also NOT "subjective and argumentative" . If something is out of your area of expertise, it does not mean that it is subjective. If you could wait enough, maybe someone at least could give some references.