The idea that the Time Lords were an (engineered) 'Higher Race' never did much for me except in the abstract, adding some 'majesty' to the tragedy of the Time War. And adding a bit of magic to the Doctor.
But I like this interpretation, this, time-awareness not being rooted in knowledge so much as a sense for it - in the Movie they show the Doctor having this sense!!! About others' timelines, which I thought was really weird. Timey wimey wibbly wobbly (Moffat'! )
I actually like....a combination of it actually - or rather - fuzziness to it. I like the more 'science/history/tool-based' and deductive/empirical/chronological nature to the Doctor interacting with timelines from their own timeline from the modern series than the Doctor getting all the scoops all the time. Fixed spots - innate sense. Flow and shape - innate sense. Pushing at time-states to get to a desired present - innate ability, but more like instinct. But the terrible awareness stretching all the way around and around - I think I'll take that as a function of the Time War.
The Time War/Lock functioning as a Great Filter in essence for all 'higher races' is that wonderful flexible and fixed time nonsense that i adore because it's so...as in always been and a paradox in itself.
So they fell prey to the very thing that made 'lesser races' even incapable of following or remembering the Time War. But to a higher dimension