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.