He rejects formalism and so there's not much to hold on to or debate. He's not the first person to say he doesn't like sets or the infinite, but since you can undisputedly work a formal theory of theories like set theory, his attack is just an ethical "we shouldn't", "math shouldn't be this and that."
The "other side" on the extreme is set theorists in academia - and they don't do any normal math and no other mathematician really knows about their ultra-large cardinals etc.
Where size issues come into play, say when algebraists try to be extra formal about treating functors as actual mathematical objects - such that questions arise like "is the domain of this object actually a set" - and stuff like that, then those will argue category theory or set theorists about set theory,
see e.g. Scholze recently here
https://mathoverflow.net/questions/382270/reflection-principle-vs-universesbut this is always just an annoyance to "normal mathematicians" (if you want to call algebraic geometers that - in any case they make for a large group of people in math depts today, even if their results are also usually just about the structures they defined 20 years prior, today.)
Other than that, professional mathematicians stop caring about the complications with the reals (as in, the properties that demand very strong axioms from them, unlike graph or number theory, and such) about the same time as most of people here do.
Those professional mathematicians heard, when they were 22, that the reals R are an ordered field like the rationals Q, but on top of that, every "number" that's definable as a limit (or lower bound of any subset) is also one.
The jump from goes unnoticed. The attitude is
"well people have been working with this since 130 years, so it'll be fine. honestly don't care about the ontological implications of the Powerset axiom and how it restricts the mathematical universe by implying a tower of other structures above me, I just want to study the zeta function."
