Since the event horizon is defined as the boundary within which the escape velocity is greater than the speed of light, and escape velocity is the speed required for that object to reach infinity away from that point, why can't light escape the event horizon even if it doesn't reach infinity away from it?

I assume this is a problem with using the Newtonian idea of escape velocity to derive the Schwarzschild radius, as although it arrives at the correct answer it describes a situation where light could leave the event horizon, even if only temporarily.