My read from looking at the literature is that the feeling is more like "black holes are a mathematical theory that have not been falsified by observational evidence, and for which we have some compelling, but inconclusive data, reconstructed from intensely noisy and poor resolution sources. Seems likely, but more data needed."
If you're sufficiently cynical, you can describe almost all astronomy results that way. Lots of alternate explanations for the observations have been proposed, and fail to match the full evidence.
Just how much evidence are you going to demand of phenomena hundreds of lightyears away before you say, "yeah, that's probably what it is"? Anyway, do be sure to read the whole list of observations linked above by csours, as there are quite a few.
I don't think it works like that?
My read from looking at the literature is that the feeling is more like "black holes are a mathematical theory that have not been falsified by observational evidence, and for which we have some compelling, but inconclusive data, reconstructed from intensely noisy and poor resolution sources. Seems likely, but more data needed."