Why does Newtonian physics give the right radius for a black hole? When computing the distance at which the escape velocity is equal to the speed of light, the result is equal to the Schwarzschild radius. Is this just a coincidence, or is there a physical reason why the Newtonian approximation give the correct result?

If this is not a coincidence, then why is it OK to use Newtonian physics here?
As explained in this answer, dimensional analysis indicates that this radius depends on $GM$ and on a velocity squared. However, the fact that the two quantities are exactly equal, rather than there being a constant factor between the two is a pure coincidence.