Yes, you nailed it. In JKR's universe she really focuses on choice. She gives us characters with similar upbringings, and shows us how it is their personal choices and not their environment that matters. It doesn't matter what you are born, it's the choices you make. Whether this is consistent with real psychology is not really relevant, I think. It's the message within her fictional universe. Although Tom and Harry really embody this a lot better than Snape and Harry, I think.

Snape's tragedy isn't his upbringing (in fact we know next to nothing about it), it's the poor choices he made and not realizing what they were soon enough.