"transfinite set theory isn't useful because any applications it has could be replaced with finitist math" would be a misleading way to phrase it. Like, analysis is not an "application" of transfinite set theory. It doesn't presuppose the existence of any transfinite numbers. No piece of applied math does. There is nothing which needs to be replaced with "finitist math".
To repeat my earlier comment: analysis on mainstream foundations uses infinite sets of multiple transfinite cardinalities, and there are results like "countable sets have Lebesgue measure zero", and the difference between finite, countable, and uncountable sums/unions/intersections, that are relevant to proving results that have practical significance, as well as (crucially) to avoiding erroneous calculations.
If you want to prove the same results in a finitist framework, there is nontrivial work to do, and few mathematicians are interested in doing it.
If your contention is that only calculations matter, not proofs, I would agree with you that transfinite set theory may not be relevant. You can do calculations without any rigor at all. But I think the position that proofs are of no practical value is untenable. It is historically simply not the case that engineer or physicist intuition is a sufficient guide to deriving results reliably.
I'm actually interested in what these results of practical significance would be. Practical, as in actually useful for technology or such.
I'm also not sure what you mean with "finitist framework". As I said, analysis existed before set theory, it doesn't require any special framework. In fact, the system of natural numbers, real numbers, and complex numbers can be axiomatized just with second-order logic. Without set theory, let alone a transfinite set theory like ZFC. And normal mathematicians wouldn't even use a formal logic here, they would just write down those axioms in plain English.
I do believe we are getting sidetracked by the example. The original argument was that some fields are studied for their own sake. And that is a good thing.
Pretending that the reason to study them is that they may prove useful in some undetermined future is just a distraction used to convince others, usually in order to provide funding or as a means to gain status. We should simply embrace that things are worth studying for the sake of knowledge itself instead of letting the pursuit of knowledge be corrupted by all devouring capitalism which reduces everything to a monetary value by denying the existence of any other kind of value.
