I don't totally buy the idea that a show needs breaks to remain wanted/interested/enjoyed by the audience regardless of quality, though it can be a factor....
Look at the original run, the 3rd and 4th Doc (early 4th doc era) were very successful, had loads of classic episodes and is widely considered to be DW at it best (classic era) and they were series 7 to 14, indeed series 13 is considered to be the best series full of DW gems and that was in the shows 13th year, and this is with no breaks.. and yet when you watch it the show doesn't feel tired... rather the opposite the 3rd doc's era and the first 3 years of the 4th doc's era feel fresh and vibrant, despite following two previous Doc' who had been on the air for six years prior.
So the notion that the show needs breaks after set periods of 3-4-5 or whatever years isn't necessarily true, it certainly wasn't the case with classic who that went 14 years delivering a vibrant and creatively successful programme that kept people watching.
The key thing was that those 14 years had the guts to really change and evolve, not just surface and actors but really change the tone vibe etc of the show, while still keeping the core essence of DW, all the while producing many well told enjoyable stories with fantastic leads.
I think the problem with DW now is that its changed it surface in these last 10 years of its revival but hasn't changed the guts underneath it all that much, just the natural change of Moffat taking over from RTD, there's only been two versions in 10 years that ultimately share many similar beats.
Then you have the quality of stories that I think has suffered under Moffats time, the show feels tired because rather then being a fun dangerous show where anything could happen it has become instead a insular self indulgent who-lore-fest where there's no consequence to anything that happens (death is cheap, earth survives things it shouldn't) and the same beats are hit again and again hence reducing a sense of wonderment and excitement about what could happen, oh the Doctor is going to die again, people are ganging up on the Doc again, time's gone wobbly again, living for a very long time again blah blah blah.
I think the fact that people think that DW needs a break shows its not working as good as it could be, the programme has many in-built mechanisms to keep it self renewed, but they have failed to diversify and put the programme on different tracks, instead they've played it safe and kept to the same track that proved to work but there is only so long you can stay on one track before people jump off.
Classic DW worked really well for it first 14 years because it was doing enjoyable interesting stories with great characters and had the guts and will to change and evolve and switch to different tracks, therefor keeping people entertained with its different flavours. After series 14 it started to produce episodes that weren't so well done, stories that won't so interesting, it became ever more creatively safer, familiar, insular and self indulgent (much like series 9).. it stayed on it's well ridden track offering the same flavour of DWness and hence fizzled out and got cancelled.... Irony being that its last series actually did start to do what the programme needed and was evolving changing track and refreshing itself, but it was all to late.
So yes eventually DW like anything can/will suffer from audience fatigue but the show has proved in its past that it can go a long time before that happens, and indeed that fatigue only happened because its makers in the late 70's and 80's failed to do what their many predecessors did so well in the first 14 years, the fact DW is feeling tired now is a direct failure of its makers rather than the concept of the show and people wanting to watch good DW.