"I don't get why we teach verifying trig identities. I can't think of a single purpose other to get students familiar with fundamental trig identities."
I think of trig identities as a time to re-enforce deductive reasoning, which is a core idea in mathematics. So it's not really about the trigonometric identities themselves IMHO, but about the core ideas that we can use equality to problem solve, and that we have to establish deductive reasoning in order to solve these problems.
I always viewed trigonometric identities as problems I had to solve where I had a limited set of possible rules, and the objective was to decide if the given statement was true or not.