Apparently there is an issue with the AOD functionality on the stock FPOS. It causes the screen pixels to not turn off when AOD is activated.
See here: https://forum.fairphone.com/t/fp5-doesnt-seem-to-turn-oled-pixels-off-when-displaying-a-black-image-video/101991/54
And here:
https://forum.fairphone.com/t/fp5-battery-life/100561/21
Since this is probably a kernel issue and I imagine /e/OS uses mostly the same kernel as FPOS, I guess the problem was also present with /e/. That probably explains why they removed the AOD option. And since the problem with stock OS is not solved yet, I guess it will take some more time before a solution comes for /e/.