But you cannot prove, formally, that formal system A talks about formal system B, without appealing to another formal system C (and then how do you know that C talks about A?). I already made this point in the OP.

If in formal system C you can proove that formal system A prooves statements about formal system B, and since

Not quite: what I have here is game formalism with the additional claim that the games are 'real' in a Platonic sense (and nothing else is).

then it means that system C is a real meta-theory for system A and B, and since B is real, then A is making 'true' statements.

Thus, while paradoxes may conceivably destroy the utility of mathematics, they still could not destroy mathematics itself.

It's the same thing. What distinguishes mathematics from fantasy narrative is the strict adherence to a set of rules: if some system is inconsistent, then it's equivalent to have no rules (that is, all inconsistent systems have the same proving power).

Good point, that should have been 'class'. Fixed.

The distinction between set and class is only meaningful in a formal system... and I don't think you want a theory able to talk about the quantity of the totality of the real formal systems...