Yeah this has always been a bit of a problem in the modern series compared to the classic one really. In the original series, he's really just a wanderer through time and space. It's basically where Arthur Dent and Ford Prefect in Hitchhikers Guide to the Galaxy came from. In the modern series they try to have it both ways - just a flippant guy who just wants to have a bit of fun, but also the last of the Timelords, the worst enemy of every monster, the centre of every big event in history, etc. I sort of get why they do it, and it makes sense in a way - most modern tv includes a bit of metatextual overlap between the pure plot and the character's personal journey, and he was involved with a lot of shit over the years (and sometimes good stuff comes out of treating him like this larger than life presence), but it also removes a lot of tension. And there's a lot of repetition in that idea.
I prefer the idea from the (especially early) classic show, where he's just a guy who happens to stumble into these great adventures. It's not all about him, he's just a guy who's there. A tourist, who at the end rides off into the sunset like Lucky Luke without anyone ever knowing who he was or that he was anyone really important at all.
When the show disappears from tv for bit and it comes back after 5 or 10 years, I kinda hope they move back to that part of it some more. The feeling you should get from the Doctor's travels isn't supposed to be a Lord of the Rings-type epic quest. It's a drunken roadtrip you make in your late teens/early twenties with a bunch of friends, full of wine, girls, weird adventures and dumb mistakes. If you know what I mean. And not that epic tales can't happen ('epic' as in LotR, not gaming-epic), it's more that that shouldn't happen so often.
Although I admit, I kinda like the President of the World bit. It's a fun little joke. But I get why you wouldn't agree with it.