Another intriguing answer came from JGWeissman. Apparently, as we learn new physics, we tend to discard inconvenient versions of old formalisms. So electromagnetic potentials turn out to be "more true" than electromagnetic fields because they carry over to quantum mechanics much better. I like this answer because it seems to be very well-informed!

I don't like this explanation- while potentials are useful calculation tools both macroscopically and quantum mechanically, fields have unique values whereas potentials have non-unique values. It's not clear to me how to compare those two benefits and decide if one is "more true."

The alternative way to look at it: if you only knew E&M, would you talk in terms of four-vector potentials or in terms of fields? Most of the calculations for complicated problems are easier with potentials (particularly for magnetism), but the target is generally coming up with the fields from those potentials. Similarly, most calculations in QM are easier with the potentials (I've never seen them done with fields, but I imagine it must be possible- you can do classical mechanics with or without Hamiltonians), but the target is wavefunctions or expectation values.

So it's not clear to me what it means to choose potentials over fields, or vice versa. The potentials are a calculation trick, the fields are real, just like in QM the potentials are a calculation trick, and the wavefunction is real. They're complementary, not competing.