When an object is squeezed to its Schwarzschild radius it becomes a black hole (made by density) and its mass does not change (its gravity doesn’t change), but if its mass doesn’t change (its gravity doesn’t change) how does light not escape the black hole? The question is not why a black hole is black, or why light does not escape black holes in general. The question is why light does not escape a Schwarzschild black hole with a small mass (which, presumably, means the gravitational pull does not change either).
The problem I think is that you are assuming that the force of gravity is given by
that is, Newtonian gravity. You are assuming that the value of $F$ on a new material will also remain constant, despite the fact that the black hole should have just gained mass. I'm unsure why you think gravity not changing is a definite proof that light won't escape, but perhaps if I can help you find a flaw in your logic, you will be able to solve it for yourself. The equation above is a useful approximation not near a blackhole. Near a blackhole, we reach what is called Schwarzchild spacetime and we must deal with things accordingly. When an object reaches the Schwarzchild radius, it does not become a black hole. It must first reach the singularity which takes some time after reaching the Schwarzchild radius. The mathematics are understandable to undergraduate students: in this sense, I just mean that even with a few university math courses under your belt, you should at least to some degree be able to follow the derivations. I suggest reading Black Holes: An Introduction the Second Edition by Derek Raine and Edwin Thomas if you are truly interested in understanding why Newtonian gravity is not a viable option here.
I think the above is a solid response to set you on your way, but the truth is simply that if you think of gravity intuitively, it usually works; but when you are discussing black holes, this intuitive approach fails.