There is no "process by which we educate any arbitrary person". Some won't succeed, no matter what portion of our resources we blow on them. The mere attempt causes our best teachers to leave the profession; few enjoy being powerless babysitters.
Your "who deserves it" terminology has a problem. It suggests that we are dishing out reward or punishment, but really could provide the best for everybody. We can't actually provide the best for everybody. This is partly because resources (including non-clonable human teachers) are limited, and partly (mostly) because the less-capable students interfere with education for the more-capable students.
I may have overstated my position. I know there are some intractable kids that can't be educated. I just don't think the success of parents is a good measure for us to be using here. Not because it isn't a good one. It is. But because it disincentives anyone without such parents from even bothering to try. They will always be measured as the child of a not-successful person. Many such "poor-yet-viable" kids become indistinguishable from the intractable kids very quickly when they become aware of such attitudes. It is far more destructive. We only need an elite in proportion to scarcity. It is in fact only when we can provide an even playing field that the elite can be elite and not merely aristocratic.
> Kids with the potential to succeed come from parents with the potential to succeed.
There's another designation. One you have a genetic incentive to profess. Shame on you. Honestly. We should improve education by... improving education. As in the process by which we educate any arbitrary person. This "who deserves it" business is a distraction that perpetuates itself.