The question of race in Doctor Who is a complex one and we've discussed it here in this forum a couple of times. I've been thinking about it again more recently while watching the Davison era which is incredibly white, yet deals with colonialism a lot. It certainly deserves its own thread.
While I can see how the role of POCs in this episode can come across as upsetting now that it's been pointed out I believe that it wasn't intended that way. On the contrary, it seems to me that Doctor Who (its modern incarnation, mind you) strives to show diversity in its cast. Due to the current Doctor and companion being white this leads to many of the supporting roles being played by non-white actors and actresses. Not all of those roles are complimentary. However, generally the show shows blacks in a variety of roles, jobs and levels of society. This episode also featured a black maths teacher saving the Doctor and Clara from being killed and the world.
The Doctor (in the new show) tends to dislike people he suspects to be his companion's boyfriends (e.g. Adam, Mickey) at first. In this case, the Doctor couldn't imagine a soldier being anything else than a PE teacher. His race didn't play any role. The dialogue made that very clear. I admit that the connection is very unfortunate, though.
At the end of the episode, the Doctor admits that he was wrong about Danny. The Doctor isn't always right and he isn't always entirely good. That's also true of the old show.
The interpretation of the Timelords as aristocracy is that of Danny and thus not necessarily accurate. However, I always thought that in the old show there was an aristocratic element to them, at least to the High Council with its titles of Lord Chancellor and such. Even on the old show there was a Guard on Gallifrey so I don't see the discrepancy between that and the dialogue in "Listen".
By the way, I feel that Danny was actually spot-on calling the Doctor an officer which is why the Doctor was so upset about it. And certainly there is a certain aristocratic arrogance inherent in being a Timelord. Ten said he was a Lord of Time in response to the King of France.
The interpretation of the Doctor running away to help the oppressed is that of Robin Hood who mixed his own story with that of the Doctor to make a point. However, I don't think that's really wrong. Surely, the Doctor's motivations to leave Gallifrey were complex. The Second Doctor argued for intervention during his trial, after all.