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."

You can just as easily move to a different mathematical structure where the gauge is "modded out", a "torsor". Similarly, in quantum mechanics where the phase of the wavefunction has no physical significance, rather than working with the vectors of a Hilbert space, we work with rays (though calculational rules in practice reduce to vectors).

There are methods of gaugeless quantization but I'm not familiar with them, though I'd definitely like to learn. (I'd hope they'd get around some of the problems I've had with QFT foundations, though that's probably a forlorn hope.)