Another PhD in maths here and I would say not to worry. It's the process of doing and understanding mathematics, and thinking mathematically that is ultimately important.

There's never been the equivalent of the 'bench scientist' in mathematics and there aren't many direct careers in mathematics, or pure mathematics at least - so very few people ultimately become researchers. Instead, I think you take your way of thinking and apply it to whatever else you do (and it certainly doesn't do any harm to understand various mathematical concepts incredibly well).