THX for the detailed documentation.
I never regarded my FP3 as a photographic device, only used it to document exhibition installations when my Sony DX100 IV was not at hand. So I didn’t really care for the pic/screen quality.
This changed completely when I got the FP6. Test shots and first exhibition shots passed the eye test with flying colours and effectively retired the Sony, although the pics are always a bit on the light side.
It’s not impossible, only Fairphone can say.
From a purely technical standpoint, one of those three predefined modes is near perfect in terms of color and tone accuracy. So this should be sufficient as it is what anyone should expect from a screen panel and from any company that sell devices with screens.
Lot of people waited long time to have such accurate panel out of box, for long years lot of panel where just jerk, and the rest couldn’t be set without very expensive probes and specialized materials, after tweaking each color, contraste, brightness, and so on, individually, hours of settings…
Now we all can have such “perfect” panel for cheap just with ONE setting, that’s great !
I don’t think so, but nothing is impossible 
First, don’t try to make your FP6 screen look like your FP3 one., wether with phone settings or with any app.
Use your FP6 screen as a target to set your other screens, specially your TV one.
You don’t imagine how fast you’ll get used to. And time will come where you’ll find “cold” screen not cool to watch 
You’re welcome 