There is a concept in classical physics called a dark star which was tossed around as early as the 18th century. The idea is precisely what you refer to - an object whose mass is so great that the escape velocity from its surface exceeds $c$. Newton himself believed light to be particle-like, and the implication of the dark star is that light particles emitted from its surface would travel outward for some distance, reach a turning point, and then fall back in.
This is not what a black hole is. The Schwarzschild radius $r_s = \frac{2GM}{c^2}$ is equal to the critical radius of a classical dark star only by a happy coincidence of dimensional analysis$^\dagger$; though the properties of a black hole may be superficially similar to those of a dark star, they are very, very different objects.
One way to think about what happens at the Schwarzschild radius is to note that at $r=r_s$, the nature of space and time are, in a sense, flipped around. The radial coordinate (which loosely describes the distance to the singularity at the center of the black hole) becomes "time-like" while the coordinate time becomes "space-like"; as a result, an object (or indeed, a photon) within the event horizon can no more avoid the singularity than an outside observer could avoid next Tuesday.
In contrast to the classical notion of a dark star, it's not a matter of having enough energy to escape from a black hole - it's just that once you are inside the event horizon, there are no future-directed timelike curves (which are the paths massive objects follow) which don't take you in to the singularity.
$^\dagger$By this, I mean that if you want to use Newton's constant $G$, the mass of an object $M$, and the speed of light $c$ to make a distance, your only choice is $\sim GM/c^2$ - otherwise the units wouldn't work out. The fact that the additional factor of $2$ in the Schwarzschild radius matches that of the Newtonian dark star's critical radius is coincidental.
