DONT KICK THE PUPPY!!!
It was a system app. Then the February 1st security update which made my password 6 digits instead of 4 (AND WONT GO AWAY EVEN AFTER FACTORY RESET - no nothing was "restored"). It also changed WebView to display at the top the regular apps list. That is how I noticed it in the first place. Only after that triggered me to wonder what the hell was happening did I go on a quest to remove the invasive forces on my device. It must have been a mistake as later unnotified changes have occurred in subsequent weeks, including it's return to the System Processes list which I only know exists because it was in the wrong place in the first place. While I can't be absolutely sure they didn't exist before hand, I'd never before been able to find and use such things as "networked apps" and "change system settings" permissions.
By the way, back to the top topic, allowing System UI (strangely also not a system app) to change system settings, seems to be the thing which both solves my problem and might have repeatedly allowed "Calendar storage" to have "contacts" permission. That's neither here not here, but something to watch out for if anyone else is trying to halt the invasion of the evil machine on OUR androids (No Google! It does not belong to you unless you're taking it by force of power, like an armed highway robber. ...and yes, you should ALWAYS ask before installing anything. )
If apps with more control than users helps stop the puppykickers from kicking the puppies

then, well -- it's a tough decision.
That kind of power is unlikely to be safe forever