On the contrary, if it switches to 4k when playing video one might argue the 810 SoC is perfectly capable of operating at 4k. That it doesn't is a choice Sony as made to preserve battery given 4k home screens are less beneficial than user-acceptable battery life.
Adding variables to supplement your argument isn't going to cut it. Go back to scientific theory and know that the only variable that should change in a fair test of battery longevity would be resolution (again, see Oppo Find 7 vs Oppo Find 7a). Clearly, adding in a more powerful SoC to power a higher resolution is only going to skew the battery data and misinform results.
So you're conclusion has clauses: resolution doesn't affect battery life so long as there other bits thrown into the mix to make it so. OK then. I'd simply say it does.
Maybe, and it wouldn't be the first less-adopted tech development we've seen. But that's the beauty of tech. Gotta keep pushing the envelope. Anyway, why not wait and see for yourself...?
The Snapdragon 810 proved to be a problem in running a HD1080P UI in:
LG G Flex2
Htc One M9
Xperia Z3+/Z4
And a number of other flagships, that is a crystal clear indication that running a 4K UI would be disastrous with that SoC.
You seem to forget that playing e.g. 4K vids is less demanding than running a whole UI in 4K. Android ramps up the CPU clock when you touch the screen, so e.g. web browsing, gaming etc... Would cause the Snapdragon 810 to overheat and throttle heavily if the whole UI was 4K, whereas when playing a video you have the GPU and CPU running the load at lower more manageable clocks as there is less interaction with the UI, and the load is generally well balanced out.
Yes, running a higher res requires a number of things to be done well,Samsung prove it time and time again. As much as I hate to admit, the Note 5 and S6 Edge+ with smaller batteries than the Note 4 manage equal battery life, why? This is due to them using an even better and far more efficient SoC, the 14nm Exynos7420. They manage to beat every Snapdragon 810 phone that runs a smaller display at HD1080P, that my friend is clear proof, that res is not the only factor.
Sony burnt themselves with the 810 the 1st time around, and had to do two things:
Differentiate (4k)
And Fix what went wrong (all kinds of heat dissipation solutions in Z5 series)
This forced them to run the Z5P the way they have. If the Snapdragon 820 proves to be capable of running 4K best believe Samsung and LG will jump on it and point out that Sony don't truly run 4K in the Z5P all the time, but the Galaxy Sx does and still manages X battery life.