Pay with Monero, set up a VPN, buy a phone specifically for the service. I doubt you can get any more anonymous than that. Cellular networks are by default monitored by governments, there is nothing a provider can do about it. But encrypting the traffic and getting a new phone should make that type of monitoring relatively useless. And if you never give your identity to the provider, they simply can’t know who you are.
You can use any redirecting extension, if it does not support FreeTube directly, just make it open
freetube://<youtube link>
.