@Rami. Thanks a lot for pointing out the other related question. @Damian. What reference do you recommend for Severi-Brauer schemes?

Thank you very much for your answer. Your answer is about flag varieties, and I am very satisfied with that, but the question was about homogeneous variety. Is there a way to fix this discrepancy?

Thank you very much for your answer and the related comments. It seems this would be an insightful example for proving that convex+rationally connected implies homogeneous.

Yes. More generally, where can I find examples of this kind?

Thank you very much for your answer! I was after an example of a disconnected moduli space of genus $0$ stable maps, with target a convex variety. So as an addendum to this question, can one easily infer from what you said an example of a variety $X$ whose moduli space of genus $0$ stable maps consists of two distinct points?