You are right that a black hole *as such* is almost impossible to detect. The black hole itself does not radiate anything much except [Hawking radiation](https://en.wikipedia.org/wiki/Hawking_radiation) which is "darker" than the ubiquitous [cosmic microwave background](https://en.wikipedia.org/wiki/Cosmic_microwave_background). A nearby large back hole, if one could withstand the spacetime distortions, would be a "dark patch" in front of the faintly "glowing" (in the microwave range) background. 

Another way large black holes betray their existence is by distorting light, even acting as a [cosmic magnification glass](https://en.wikipedia.org/wiki/Gravitational_lens) when the stars align just right. But this effect is visible for large black holes only and needs scrutiny to detect.

Because a black hole *as such* (and the reason for this repeated caveat is below) is practically invisible there is even speculation whether there is a [much larger number of them as generally assumed](https://news.yale.edu/2021/12/16/black-holes-and-dark-matter-are-they-one-and-same).

The reason I say that black holes "as such" are invisible is that there is a class of black holes which make their existence unmistakably known: *They are [quasars](https://en.wikipedia.org/wiki/Quasar),* the brightest objects ever, by far. They spit out radiation and matter at relativistic speeds over many lightyears, enough to disturb their host galaxies and put their imprint on entire cosmic neighborhoods. You are right: The light does not emanate from within the Schwarzschild radius, that would be impossible. It emanates form the extreme conditions *around* the black holes, which fuel extreme interactions with and between matter in its vicinity. The black hole is a bit like your blender when you forget to put the lid on: It ruins the entire kitchen, the more is in it, the merrier. The blender *as such* isn't doing anything, really ;-).

But when it has things to shred, it becomes *very* visible.