I think the video was mostly just trying to show some of the underlying workings of how a black hole could be made, not necessarily how they are made.
As for a singularity at the center of a black hole, if one was to form, when would it? Beyond the end of eternity isn't a real answer in the known universe. Likewise, trying to define what is or isn't beyond the event horizon, from our perspective, almost seems like a false dichotomy. From our perspective, that occurrence never happens, yet there are black holes. If black holes evaporate, then they would seem to evaporate in an instant at the event horizon from that perspective. Either way, whatever is inside of a black hole would seem to be "gone;" it's just a matter of when it exactly "leaves."![]()
If Hawking radiation exists, then that emitted by stellar mass black holes is so incredibly tiny that only one or at absolute maximum two particle pairs will be ejected over the age of the universe. Once the Hawking temperature is below the cosmic microwave background temperature, the black hole is stable; the smallest stable black hole is around 0.01 Earth masses I think
I do think that the video does explain how a black hole could be formed after supernova, but the example they give is incorrect.