Given how strongly you feel the burnout, and how big of a dealbreaker it sounds for you, I suspect the benefits of dodging all that would outweigh any compromise in skipping.
At the end of the day:
(1) 1-3 is a good grounding in the series and characters and a complete story in its own right,
(2) Time War is designed to be accessible to newcomers,
(3) 4-6 is very much a side quest, and
(4) you can always fill in the gaps later.
Initially I skipped 4-6 and jumped straight to Enemy Lines / Intervention Earth. While it’s undeniably a detour (an interesting new direction they sadly
had to backpedal on against their original intentions), I enjoyed it as a palate cleanser, and it flows into Time War nicely. I honestly can’t remember anything that would make it required listening before Time War, though; it’s more
tonal continuity than anything else.
The Brigadier raises a good point: there are some character developments during 3-6, mainly with Narvin, that add weight some of his decisions in Time War. Nothing you’d notice if you weren’t in the loop, and nothing I was too upset to have missed when I caught up afterwards, but it probably would have given me more appreciation of his character in the later sets.
Would it be helpful or cruel if we dropped two key lines of information here behind a spoiler tag?