There is a distinction between logic, which is just the study of formal systems where you can prove theorems as in any other area of mathematics, and the philosophy of maths and particularly of mathematical logic. The same applies for set theory.
The first one can assert that (in classical first order logic) e.g. there are uncountable models of the natural numbers; this is irrefutable. The second one asks things like "is classical first-order logic even true and/or adequate in an epistemic sense" (this is different from the more pragmatic question of "is classical FOL useful for the problem at hand")? Similarly, something like Gödel's incompleteness theorems are unequivocally true but the question of what they "mean" deep down is nothing that really affects mathematicians' work in general.
Irrefutable and "unequivically" true if you take some classical FOL as a productive method of producing knowledge, a philosophical assumption. Philosophy came first, mathematical logic is just a formalization of that and the reason it was formalized at all and not somethimg else is epistemological.
Your entire argument attempting dismiss philosophy is philosophy.
I said that as soon as you fix some axioms (such as those of classical FOL), the conclusions are irrefutable. This is where mathematics begins. The question where those axioms come from or whether they are "true" are philosophical. The two disciplines are related, but separate.
Lots of mathematicians have different "foundational" beliefs from each other; some have studied or deeply thought about the philosophy, others may just speak to their intuition. However, this doesn't change the fact that they all come to the same conclusions from the same premises. E.g. a constructivist wouldn't be able to claim that a classical proof is "wrong", only that it's non-constructive and therefore unacceptable for some (philosophical or practical) reason; in fact non-constructive proofs can be seen as constructive proofs of some meaningless strings (e.g. the constructivist will maybe dispute the fact that there are discontinuous functions, but they will certainly accept the existence of a first-order derivation of the string representing "not all functions are continuous" from the axioms of set theory), the constructivist would just dispute that there is any meaning to these strings...
The first one can assert that (in classical first order logic) e.g. there are uncountable models of the natural numbers; this is irrefutable. The second one asks things like "is classical first-order logic even true and/or adequate in an epistemic sense" (this is different from the more pragmatic question of "is classical FOL useful for the problem at hand")? Similarly, something like Gödel's incompleteness theorems are unequivocally true but the question of what they "mean" deep down is nothing that really affects mathematicians' work in general.