I have a similar problem with Apple TV videos suddenly stopping or the service not being available. All other streaming services work flawlessly at full 4k. It took a bit of digging since my home setup is not typical. I run Server 2022 Active Directory with Windows 2022 DNS and a Sophos Firewall (I am a bit security minded). I also run DNSSEC and everything TLS.
I can reach every website on the Net without issue aside from some shady looking ones that cannot be validated via DNSSEC. All streaming services work flawlessly, except Apple TV+. It will work sometimes but then suddenly stop. The Apple TV servers are not DNSSEC compliant (Verisign DNSSEC tests confirm this).
My Sophos Firewall also has a DNS server running, but it is not robust so I do not use it for my network. The Sophos DNS server does not support DNSSEC. When my Apple TV+ fails to connect, I point it to my Sophos DNS server and it starts working (again no DNSSEC). The interesting part is I can no longer access Disney+, Prime or other streaming servers without DNSSEC. So, I need to switch back and forth manually changing the DNS server on my Android box. I have the same issue on all my other devices including my iMac.
If I wait a 1/2 hour or so, Apple TV+ will start working again for a period of time. It will then stop again after another period of time unless I point it towards the DNS server without DNSSEC.
If you are streaming from different networks, perhaps some are using DNSSEC which breaks Apple TV+ access.