I'm sure most theoretical physicists would call themselves mathematicians. I don't agree physics should favour creating testable predictive models, because who are we to say what's testable in the future? Take the theory of relativity, for example. Would you have stopped Einstein et al from researching it, just because it wasn't testable at the time?

In my opinion, physics is and has always been about answering the big questions, the how's, the why's of our existence. That new technology sometimes comes out of it is a bonus, not the ultimate goal.