The thing is, the show itself almost never calls him on his shortcomings. The Master stuff is fridge logic that no one in-universe ever seems to quite put together (I understand that originally, the Master was supposed to mock the Doctor about this, but it never happened, so doesn't count.) And if they do, they certainly don't seem to have the good sense to be outraged because of it. His failings tend to get either swept under a rug or used as fuel for his romantic persona. He's portrayed as somehow "better" because he's a Byronic twit.
What are we told about him though, what's reinforced in episode after episode? "He's like fire and ice and rage. He's like the night and the storm in the heart of the sun. He's ancient and forever. He burns at the centre of time and can see the turn of the universe. and... he's wonderful."
He's a "lonely angel" or a "lonely god" depending on who you ask.
Companions and incidental characters like Jackson Lake in The Next Doctor talk him up every chance they get. It's obnoxious and schmaltzy.
Waters of Mars was probably the only overt example of an episode where 10 is called on his shit in-universe.
Jack gets in a few digs in too at the end of time, but with a wink and a smile. Comes off as playful banter rather than a genuine expression of frustration.
Interestingly, I feel like since the start of the Moffat era we went from a false-modesty doctor surrounded by fawning sycophants to openly narcissistic Doctor with long suffering, eye-rolling companions. Neither is really "better" than the other, simply a matter of taste. Personally, I'd rather the Doctor had a high opinion of himself while his companions are less convinced. Seems more natural to me.
The exchange that sums up Moffat era companions for me.
Rory: He'll be fine. He's a Timelord.
Amy: That's just what they're called. It doesn't mean he actually knows what he's doing.
