This is a nice overview that gets technical but what I particularly enjoy is the comments. As this is just above my range of comprehension, I go back to this page on occasion when I understand another piece of it.
Yes, I think you’re correct.
“Gauge symmetry is required in order to make quantum electrodynamics a renormalizable theory, i.e., one in which the calculated predictions of all physically measurable quantities are finite.
Gauge symmetry is closely related to charge conservation. (In physics, charge conservation is the principle that electric charge can neither be created nor destroyed. The net quantity of electric charge, the amount of positive charge minus the amount of negative charge in the universe, is always conserved..)”
Tying theory to observation / measurement is also, more generally, what keeps this all in the realm of “a science”.
Without it, it could easily become mathematical toy, a fiction.
There was a conference a few years ago about this with some calling to place Theoretical Physics into the Humanities and no longer in the Sciences.
It didn’t come to that, thankfully.