Does math always predict physics? Diracs equation predicted anti-matter and SU(3) symmetry additional baryons. Einsteins GR equation has oscillatory solutions. But no experiment has seen gravity waves yet.
So, it's too easy to assemble a small narrative of history that gives this impression, and forget every other model that people crated trying to get to those ones that hold.
It's not exactly that math always predicts science, but that math allows us to filter theories based on what we else we know to be true. If working out the math based on your theory gives us answers that don't match reality, then there's probably something wrong with your theory.
The period of their orbit is decaying over time, at a rate that is in precise agreement with the predicted energy loss due to gravitational radiation in such a system (the observed rate of energy loss is 99.7% +/- 0.2% of the gravitational wave prediction). So while we haven't seen them directly yet (they're really, really hard to detect!), there's very little doubt that they're out there more or less exactly as Einstein predicted.
For example with GR, it explained for example the precession of Mercury's elliptical orbit, and predicted time dilation (used now in GPS for example) and bending of light. It is true that gravity waves remains to be directly detected.
This is how physics have usually progressed in the past: Have an old theory, have some experiments not explained by that theory, invent new theory that can be reduced to the old theory and explains these new experiments and give new predictions. (The subject of string theory is somewhat different however.)