Do I need a good understanding of mathematics in order to excel at linguistics?

IMHO: for most disciplines no.

Steve Rapaport has spoken on Applied Linguistics; but Applied Linguistics is a very different discipline to Theoretical/General. Phonetics is an experimental science, so you’ll need statistics there.

Reconstructing in historical linguistics requires a degree of rigour and thinking in terms of rules which is a bit like maths, but only a bit. Ditto phonology and syntax, and I guess morphology. The other branches not so much.

Formal semantics has much in common with mathematical logic. But you’ll get more out  of philosophy than mathematics if you go that way.