Your argument is exactly the one John Mitchell gave in a letter to Henry Cavendish in 1783 about the concept of dark stars, see [wikipedia](https://en.wikipedia.org/wiki/Dark_star_(Newtonian_mechanics)).

I would classify this as an argument that works because of dimensional analysis. You have all the right basic physics, so you know your fundamental parameters are $G$, $c$, and $M$ and you need to end up with a length. Any logical derivation working with $G$, $c$, and $M$ that results in a length scale has to get the Schwarzschild radius as the answer, up to a dimensionless factor. By luck, this dimensional analysis argument gets the right dimensionless factor in this case.

There are other contexts where naive arguments about gravity without using GR get the wrong dimensionless factors. For example, [the bending of light in a gravitational field](https://physics.stackexchange.com/questions/425890/does-special-relativity-newtonian-gravity-predict-gravitational-bending-of-l). Because of this, I don't think it's possible to think of GR as simply a consequence of a simpler and less complete theory of physics, like Newtonian gravity or special relativity.

That's not to say that GR can't be understood as a logical consequence of deep physical principles. There are arguments that GR is the inevitable result of special relativity and the equivalence principle (at least at low energies), and the equivalence principle follows from having a force exchanged by a spin-2 boson. But, while these arguments are beautiful and deep, they rely on extra information (or assumptions) not already implied by special relativity or Newtonian gravity.