Finally my phone updated to 21.05.0!
I used TWRP as explained here #twrpwoflashing to access my phone data on the computer and save it there. (Doesn’t have anything to do with the update, but at least I felt more comfortable trying again, knowing my data is save).
Then I tried to flash the images manually as linked above - that also failed, but also didn’t do any harm.
At a random attempt to reinstall 19.11.2 manually, that eventually worked.
So, I dared to do the update to 21.05.0 with the fairphone updater again, it exactly went like @scribe writes here: FP2 not rebooting after updating to 21 / Android 9 - #15 by scribe
So, my phone runs smoothly on 21.05.0 for 2 hours now, all data still there. If you experience the same problems: Really just keep trying, try different cables, USB ports, computers if available - it was worth the patience! Feels like having a brand new phone right now after 5 years FP2 
Thank you for your support!