I don't think the singularity itself "actually exists" but is more just a mathematical concept we use to explain the phenomenon. Relativity comes into play extremely severely with black holes and the time dilation essentially approaches infinity at the event horizon so that from the perspective of the distant observer, namely us, it takes an infinite amount of time for something to cross it. And infinities can work well with mathematical functions but not so well irl.

I don't think the matter lives on the surface so much as the core of the dying star essentially gets so much mass in such a small area that it "freezes" more or less in a state of collapse just a infinitesimally small degree of density away from what would be necessary for a black hole to "actually form" to us and anyone else outside of it, it's functionally identical to a black hole. To anyone falling into one you'd eventually catch up to the timescale of the collapsing star and get vaporized in the subsequent explosion and spat out across the universe as light and heat, just redshifted out to near infinity. This fits with my understanding of the Hawking Radiation theory as well though I don't well understand the specifics of it. But it seems like a black hole will slowly lose mass at an incredibly slow but accelerating pace until it gets very small then releases the last of it's pent up energy much more quickly and finally in the last seconds of its life it will release the last bit of energy extremely quickly and "explode" leaving no black hole left. This process however basically takes such a long period of time though that it might as well be infinite.

So it seems to me that black holes are basically stars that are in the process of dying but they died so violently that a small part of them essentially got blasted across time to the distant future so far that it's ludicrously insane to think about, kinda like how if s rocket explodes at high speed all.its parts are strewn out across continents, but like with weird relativistic effects thrown in. It's like the universe can't handle the energies involved and freezes until they are dissipated incredibly slowly across time. And the "black hole" we see is basically just a hole in space time left by the explosion of the star. Or rather almost a hole, but not quite. Almost a singularity, or a collapsing star that is ever progressing towards that state but never quite reaching it, and ultimately never reaching it since you can't travel faster than light, and you need to actually make a real singularity.