I think there is a fairly straightforward answer to this question coming from reverse mathematics. According to Wikipedia, "finitistic reductionism" is represented by the system WKL${}_0$, and reverse math people know a lot about what mathematics can be done in WKL${}_0$ and what mathematics requires substantially stronger systems. Whether WKL${}_0$ corresponds to finitism exactly may be debatable, but in any case we know a lot about what can and can't be done around this level.

To answer the question directly, I think it's fair to say that most ordinary mathematics can be done in WKL${}_0$, and nearly all of it can be done in ACA${}_0$, which is predictive if not finitistic (although full predicativism goes substantially further than this).

I think there is a fairly straightforward answer to this question coming from reverse mathematics. According to Wikipedia, "finitistic reductionism" is represented by the system WKL${}_0$, and reverse math people know a lot about what mathematics can be done in WKL${}_0$ and what mathematics requires substantially stronger systems. Whether WKL${}_0$ corresponds to finitism exactly may be debatable, but in any case we know a lot about what can and can't be done around this level.

To answer the question directly, I think it's fair to say that most ordinary mathematics can be done in WKL${}_0$, and nearly all of it can be done in ACA${}_0$, which is predicative if not finitistic (although full predicativism goes substantially further than this).