If a black hole itself emits no radiation, what evidence do astronomers and physicists today have that the theory of black holes is correct?

Einstein's general theory of relativity is the theory behind black holes has been tested with a wide range of experiments of which all confirm the predictions the theory makes. We cannot see black holes phenomena inside the event horizon, we do observe things outside this limit.

Black holes in binary star systems leave signs of their presence on neighboring star thats detected and the signs include X-ray emissions, accretion disks, and large orbit perturbations.

this is the evidence that astronomers and physicists have to show that the theory about black holes is correct.