If he does something impressive, or makes a good argument, they laugh at him and feel good about themselves. "Both of Harry's parents howled with laughter at that, like they thought it was all a big joke" (Ch. 2) is their reply to Harry's description of school as "child conscription".

No, this is their reaction to Harry's explanation of why he isn't in school. He says that he's a conscientous objector to child conscription. They find this very funny, because the reason they took him out of school is (possibly among other things) that he bit a math teacher. This is not Harry doing something impressive or making a good argument.

What follows feels almost like a fully-general criticism, so I don't know how seriously you should take it:

This post, and the comment thread, feels to me very much like you're arguing towards a bottom line, like you've decided what personality traits you're going to find and now you're looking for things that you can present as evidence towards them, if you squint and don't read the text. Another example would be

Feels entitled - has unreasonable expectations of special treatment? Check. Harry requires subservience from the school administration, and special magic items such as the time-turner.

Harry's reaction to getting the time turner is along the lines of "what the fuck are you thinking?" Later he grows attached to it, but the initial reaction doesn't feel at all like it fits.