Are there actually theorems from which you take those explanations or are these just plausibly sounding hypothetical explanations?
Well, a model that is aware of this symmetry only has half as much data to look at and one less thing to learn.
But that's only half of it. Truth is, assuming symmetries works pretty well even if the assumption is wrong. Why? Generalization. A model with less data will generalize more (better is perhaps debatable, but it will definitely generalize more)
This is the basic idea behind "geometric deep learning". There's loads of papers, but here's a presentation.