I too miss the Doctor being part of this society and species of similar-ish beings. I distinctly remember Romana's snappy little put down when they met:
Romana: "Well I did graduate from the academy with a triple first."
Doctor: "I suppose you think we should be impressed by that."
Romana: "Well it's better than scraping through with 51% at the second attempt"
Doctor: "That information is confidential!"
See though, the fact that there are a society of his peers that are as intelligent as the Doctor is, and as (or usually more) educated, doesn't actually detract from the Doctor's specialness. Because they're mostly stuffy old academics interested, inward looking, chronically prone to inaction. The Doctor
is special not because he's smarter than them but because he is more bored and so goes out and seeks experience. We see this when Romana first joins him. She is unquestionably more educated than he is and arguably may be more intelligent. But she lacks his unorthodoxy and real-universe experience (to begin with). So she has an arc as well. It's fun to see her adapt and she quickly comes up to be an equal partner to him. In her own very delightful and playful way.
The Time Lords as written are much like the wizards in Terry Pratchett's Discworld novels - lots of power and knowledge but just institutionalised and rarely looking outward into the world. And it's probably no coincidence that both franchises are British because in American culture power is expected to be associated with the use of that power. You think of presidents and wall street kings... but in Britain the hidebound, sleepy establishment, royals, etc. were a thing that both the Who writers and Pratchett were likely familiar with. There's no more petty and self-involved politics than Academic politics.
That's why they had to bring back Rassillon to lead them in the Time War. Yes, he's a genius, smart even by their standards. But the real quality he brought was he was from their past as a species of active, ambitious people. It's also why they turn to the Doctor to do things. None of them want to risk their own necks, half of them think it's beneath them to get involved and privately, nearly all of them probably think the Doctor with all his experience and insane risk taking (from their point of view) will do it better than they could.
I think the Time War and loss of the Time Lords was a good call to begin with. It gave Ecclestone and Tenant some real emotive drive behind their characters that they did great work with. Eleven too, in a different way ("the man who regrets and the man who forgets"). It led to really dramatic moments like Martha telling the Doctor that Professor Yana had a chameleon watch and the frantic race to him before he opened it (unsuccessfully). But some of the later handling was really bad. I particularly disliked making them this semi-evil group that we see by the end of Capaldi's run. Though I actually liked their intervention with Eleven when they granted him the new regeneration cycle.
For all that Hurt is a brilliant actor and he did well with the role, it should have been Eight in the 50th anniversary. He should have become the War doctor.
And while I'm sperging about Time Lords and regeneration I'll just give my two takes on both. Firstly, I like how the rebels and exiles from Time Lord society have a defining name. If you notice, those still part of Time Lord society have names. The rebels/exiles are "The Doctor", "The Master," "The Rani"... there's also a reference to "The Corsair" at one point. I like it. The second take is that Regeneration makes great sense to me. For a species that lives along as the Time Lords and has the potential for such stagnancy, regeneration shakes things up in their lives. I imagine in their society proper, regeneration is an orderly and prepared for event. The Time Lord is 800, maybe 900 years old. Their body is wearing down, their set in their ways and career. The last couple of centuries have pretty much been the same year to year. They talk to their friends, say goodbye, if they have a partner they don't know if they'll still want to be wed to that partner afterwards or vice versa. They go somewhere quiet, close their eyes for the last time and - unlike the Doctor - meditate and try to guide their new regeneration. The thing with regeneration is that they keep their memories, they retain core values. But the box gets shaken up and that's a good thing.
Regeneration as a concept has always fascinated me. To take what you've learned, to retain your values but to just rearrange things and say: "Okay, this version of me has had its run. Lets see what a different approach can do with these things."
