Robert Chase and Tall Poppy Syndrome I find the take in the fandom that "Chase doesn’t care" or is just "House’s lapdog" super interesting. As an Australian, I think Chase's reluctance to challenge House and the rest of the team - or make any attempt to 'stand out' is reflective of Tall Poppy Syndrome.
Tall poppy syndrome refers to how socially within Australia, individuals who are perceived to be too ambitious, successful or self-congratulatory are criticised, or 'cut down' by their peers. In many circles throughout Australia, being a tall poppy much isn’t just discouraged, it can make you a target.
This puts Chase in direct contrast with the other fellows, particularly Foreman (my favourite frenemies, haha), who is openly ambitious and determined to prove himself. Chase, on the other hand, actively avoids drawing attention to himself. Even before working for House, he literally chose seminary school - about as far from a high-powered medical career as you can get. And then there’s his dad. Having a famous father that the show goes out of it's way to imply pulled strings to get him into House’s fellowship would only make Chase even more inclined to lay low. When you already have a leg up, the last thing you want is to give people more reason to question if you deserve it. Don't even get me started on how Australia’s work culture (Heavy, heavy, heavy emphasis on work-life balance) would influence his character. TLDR: It’s not that Chase doesn’t care. It’s that he knows exactly how dangerous it can be to stand too tall.