A singularity is a mathematical concept. Many things can be treated mathematically that are impossible in a physical world. It may be just fine to treat the 'gravitational well' as a singularity for mathematical calculations, but physically, an 'infinitely dense' material occupying zero space makes no sense.
This article is the first I have ever seen that actually estimates the real size of black holes. Most others treat them as "infinitely small". Whether something that's millions of light years away is the size of a golf ball, or or the size of the earth, or even infinitely small, is irrelevant for mathematical analysis. That doesn't make it zero size, though.
Is the 'gravitational well' a physical 'singularity'? I think not.
The first or one of the early Hidden in Plain Sight books postulates that gravity is not exactly the equation we use now, the equation that results in a “singularity.”
It postulates that at extreme levels, gravity “balances,” pushes back on itself, and that the mass of a black hole is not concentrated at a point, but is (for want of a better visual image) a hollow sphere at the event horizon.