I am becoming increasingly convinced that Wildberger's views are, if a little bizarre, at least not hopelessly inconsistent.
When I was reading the comments in the video following (MF17), somebody said something that shocked me a bit, because I was unable to give a rebuttal that I found satisfactory:
The reason I consider [the axiom of infinity] restrictive is that it forces you to accept the existence of a certain set that has all sorts of bizarre properties, like the existence of one-to-one and onto mappings from some sets to proper supersets. That screws up your attempts to assign cardinalities to sets (see the continuum hypothesis), and it doesn't even buy you much: can you have a set of all sets? Why not? The real question is, what do you think infinite sets actually buy you in terms of reasoning power?
The point about the continuum hypothesis is arguably meaningless; as I understand it CH says that it is hard to assign to each cardinality a set, but nothing about the difficulty of assigning each set a cardinality.
However, the core of the question is a bit harder for me to answer. My immediate response is it that allowing infinity gives a sort of completeness. But as the person pointed out, it doesn't finish the job, it just (dramatically) kicks down the line where the new notion of "too big" is. Introducing classes pushes it even further, but the same problem arises, I think: there is still no class of all classes.
So my next idea was, well, infinity doesn't buy me reasoning power but it does provide a satisfying source of many examples. But I'm not sure if that's true either: Wildberger's alternative to ZFC is (if consistent) a type theory, or at least it uses the language of type theory. I know very little about type theory so if you want to reference it in an answer, it would be great if you could use small words :)
With type theory on your side it's not even clear that I have a much less restrictive universe of objects which I can speak about; just a not-entirely-arbitrary boundary where sets are no longer permitted and types must take over. This could be dramatically the wrong picture, since as I said I am very new at this.
And now I'm stuck. Can anyone save Cantor's paradise for me?