This very matter is discussed in depth by Mathias, in Chapter 9 of his The Stength of Mac Lane Set Theory https://www.dpmms.cam.ac.uk/~ardm/maclane.pdf. There he shows that to prove "for all $n$ there exist $n$ pairwise nonequinumerous infinite sets" requires unbounded Separation, and to prove "there exist infinitely many pairwise nonequinumerous infinite sets" requires Replacement. Whether or not these assertions count as ordinary mathematics, I find them considerably easier to grasp than Borel Determinacy.
Meanwhile, some of the motivation of this question resonates with mine in posing these questions:
When must it be sets rather than proper classes, or vice-versa, outside of foundational mathematics?
Can one exhibit an explicit Kuratowski infinite set without invoking Replacement?