I was able to solve my Chromecast-related problems following a hint from @tinu1212 in Fehlermeldung bei Verbindung von Plex Android App mit Chromecast
I just had to add “plex.direct” as an exception in my router’s “DNS rebinding protection”.
You can find some information regarding “DNS rebinding” issues in https://support.plex.tv/articles/206225077-how-to-use-secure-server-connections/
Exemplified, for my “Fritz!Box” router there is a German tutorial how to solve these issues at https://avm.de/service/fritzbox/fritzbox-7590/wissensdatenbank/publication/show/663_DNS-Auflosung-privater-IP-Adressen-nicht-moglich/
