Okay the issue really stems from trying to enable aptX support, which was only available with legacy audio stuff.
Because if I remove/reset the overrides (of pipewire-pulseaudio wireplumber), it works. I only have A2DP/AAC codec then, which makes it impossible to use for input and output of audio (as the bandwidth is way too small).
Thankfully I found a solution using the new wireplumber, because rpm-fusion now apparently has a wireplumber-aptx plugin that is even easier to use than before! No overrides needed anymore!
After you have added rpmfusion as usual, it is as easy as doing:
rpm-ostree install pipewire-codec-aptx
That’s it! Now you can use aptX.
(That said, I can still not use the headet as an input device at the same time, as the codec then automatically switches to very ugly HSP/HFP/mSBC but at least it works as it worked in F35, great!)