I loosely identify with the schools of intuitinalism/construtivism/finitism. Primary idea is that the Law of the Excluded Middle is not meaningful.
So yes, generally not starting with ZFC.
I can't speak to "truth" in that sense. The skepticism here is skepticism of the utility of the ideas stemming from Cantor's Paradise. It ends up in a very naval-gazing place where you prove obviously false things (like Banach-Tarski) from the axioms but have no way to map these wildly non-constructive ideas back into the real world. Or where you construct a version of the reals where the reals that we can produce via any computation is a set of measure 0 in the reals.
I don't understand why you believe Banach-Tarski to be obviously false. All that BT tells me is that matter is not modeled by a continuum since matter is composed of discrete atoms. This says nothing of the falsity of BT or the continuum.
All that BT tells me is that when I break up a set (sphere) into multiple sets with no defined measure (how the construction works) I shouldn't expect reassemlbing those sets should have the same original measure as the starting set.
Won’t the reals we can construct by any computation be enumerable? What measure can they have if not zero?
Yes, they have measure zero. So the question becomes whether "measure" is a useful concept at all. In my opinion, no, it is not. It's just another artifact of non-constructive and meaningless abstractions. Many modern courses in analysis skip measure theory except as a historical artifact because the gauge integral is more powerful than the Lebesgue integral and doesn't require leaving the bounds of sanity to get there.