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.
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.