Live transcoding and playing at the same time is too much for the RPI. The same media all play fine on native players, without a hitch. Transcoding should not be necessary at all. The issue only occurs due to the browser not supporting the media type. <rant>Making the default client browser-based is simply stupid in our diverse age for a topic dealing with audio and video!</rant> Thread continues.
I've installed the "official Jellyfin Media Player". This is just a browser embed, i.e. not better at all. In fact, it's worse, because it embeds Chromium which supports even less media types than Firefox. Through "Flatpak", I've installed the old QT Jellyfin Desktop, Delfin, Fladder and Switchfin. All have native players, good. Two of them don't even start. The two other crash after playing thirty seconds of video. Great, exactly what I expected from a monster like "Flatpak". Thread continues.
The best option so far seems to be "Jellyfin MPV Shim". This simply launches an external MPV instance to play the media natively. Requires some configuration and IT understanding which not every user in the family may have. If I just set it up and leave it on, could work. Though still looking for a better alternative. Suggestions welcome!