Yep! But you have to prevent it from reaching the internet one way or another.
As for music, that's what I do but I think it's a shame to turn on the TV just to listen to music on the soundbar, so most of the time I just connect to it in bluetooth with my phone, even though it's not lossless that way.
I have a LG G4 which I connected to my non-internet access point. Home assistant accesses the API locally and the TV cannot reach internet. I only use the API to check if the TV is on but I can change the source, volume and launch apps.
I use a Vero V as media center, this one is connected to the main access point but I trust it and I need it to access my NAS.
My soundbar is offline but has CEC and eARC.
The Vero turns the TV on and off with CEC, then the TV turns the soundbar on and off with CEC and the sources of both are setup everytime.
Ça dépend du matos sur lequel tourne ton serveur, perso c'est sur un vieux PC portable et j'ai configuré le BIOS pour qu'il démarre quand le courant est branché. Du coup ça se comporte comme un raspberry pi.
SSL certificate could leak subdomains depending on how it's configured. Otherwise I wouldn't be surprised if Amazon sold this info within 20min