This worked for me. I did use the wizard eventually and that is working for me.
OP here…
Today I got a replacement FP4 and I wanted to try if it would actually work without any magic. I first used the phone for some 8 hours without Google login and everything was working just fine. Then I did a factory reset and migrated my account from my Pixel 4a to FP4 and immediately it stopped opening the screen. So this confirms my assumption that this was not a HW issue. Probably your are right and it has got something to do with the (Pixel) settings migration and ambient display.
I guess it’s time to play some adb games 
edit. I also reported this thread to the support on my open case
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Hi I just had the same problem with my new FP4. I imported my old account from my pixel 2. The solution was simply go to settings on my pixel 2, turn off always on display and make a new backup. Now it works fine 
now the question arises for me:
Is this a bug that FP needs to fix?
Or is this not rather a bug that google has to fix?
Because, it is already arrogant and cheeky to simply put this configuration option in the backup and rely on the fact that you only restore it to the same device or to a device that at least supports AoD.
But there are still many other manufacturers…
Or google itself knows nothing about this problem?
But again, I see the error actually in the creation/restore of the backup, not when restoring it.
There must be a script that checks whether the new device also has an OLED or not.
And that’s google part. Part of AOSP. Not FP
I ended up installing this as @n3utrino_ch suggested and it worked without using adb.
So I can confirm it’s a solution to my problem at least. 
There must be a script that checks whether the new device also has an OLED or not.
I’m not really aware of how porting Android to devices works but I’d assume there are some device drivers FP people are implementing, so I’d say it’s quite difficult to know who to blame here and of course blaming anyone doesn’t help. I just hope there’s an official fix soon.
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