***SOLVED... kinda***
You can find Bosch 2020 Dealer Manual
here.
On page 201 there's info regarding new mount for Intuvia (see picture). When a company decides to change the production of a component "to improve contact quality" you can be certain it's simply because the old design didn't work well... for some people
This got my hopes high.
The guy at the service shop first opened the motor cover and plugged in new mount along with new cable directly to motor and nothing improved. Then he replaced Intuvia mount without replacing the cables (as recommended in the manual) and... nothing improved
The bike still wouldn't turn on most of the times using button on the display. I can't tell if this fixed mid-ride cut-offs because these happen rarely and almost without any pattern.
I rode for half an hour on a VERY bumpy terrain outside the shop and everything seemed fine, no cut-offs, but bike still wouldn't power on every time using power button on the display. Maybe cut-offs were related to the mount, yet they are so random that it was hard to tell.
Then he gave me another (used) Powerpack 500 battery to try out. Yes, he was an extremely nice guy and determined to find the culprit. And suddenly... everything worked perfectly
The bike turned on every time using display button, no matter how much we tried to "abuse" it. I left my battery at the shop and we agreed that I try this one for a few days.
During one-hour ride home (~17km) everything was fine.
The bike turns on every time, whether I use battery or display button. I'm truying my best to torture-test it and it works flawlessly.
Cut-offs are still to be tested more. Maybe, just maybe, these two problems are unrelated, but I highly doubt it. For now my unfortunate solution seems to be faulty battery.