I can't speak for other answers to the linked question, but for my answer I was referring to most of mathematics as is currently accepted as of today. To make this precise there are two points that we need to clarify. In my answer, I linked to a precise definition for interpretability, so that it doesn't matter what specific language we are using. This is important because one could otherwise argue that mathematicians who do not write in the language of set theory (like many in history) are not actually dealing with sets. The second important way one can make the idea of currently accepted mathematics precise is to specify precisely a collection of theorems. For example we have Harvey Friedman's Grand Conjecture:
Every theorem published in the Annals of Mathematics whose statement involves only finitary mathematical objects (i.e., what logicians call an arithmetical statement) can be proved in EFA. EFA is the weak fragment of Peano Arithmetic based on the usual quantifier-free axioms for $0, 1, +, ×, exp$, together with the scheme of induction for all formulas in the language all of whose quantifiers are bounded.
In this post he further adds: "sometimes with qualifications that the article not be written by people referring to themselves as logicians". I think the general consensus is that there are likely to be only few counter-examples that are not contrived. Note that EFA is far weaker than ACA (which I was referring to in my linked answer). Also, the Wikipedia article mentions some natural mathematical problems where the answer cannot be proven in EFA, such as the optimal asymptotic complexity of a disjoint-set data structure, in this case because it involves the Ackermann function, which grows faster than any provably total function in EFA.
The reason why I single out ACA is because there is a natural correspondence between arithmetical sets (namely sets of natural numbers definable by an arithmetical formula over PA) and oracles for finite iterations of the halting problem. It may be a surprising that this is all we need to encode a vast portion of ordinary mathematics, which according to Stephen Simpson refers to "mathematics which is prior to or independent of the introduction of abstract set-theoretic concepts, [including] such branches as geometry, number theory, calculus, differential equations, real and complex analysis, countable algebra, the topology of complete separable metric spaces, mathematical logic, and computability theory". In contrast the Grand Conjecture is restricted to arithmetical sentences, because real numbers cannot be encoded in EFA.