Yes, confirmed it now works again on Firefox Mobile. Today was the day where they switched over the domain, so I guess that makes sense.

Nah, a private instance works fine. Public instances see a lot more traffic and if they use their public IP address for accesing YouTube it is trivial for YT to ban them. If you want to host a public instance, you should use a proxy or VPN for all the YT traffic so yo can easily change the IP address.