This has been reported in many places for weeks. What I've gathered so far is:
- This happens after update to iOS 16.1 (and then iOS 16.1.1 also)
- There are inconsistency as some video does play. (To me it seems like ones with ad is the one that doesn't play.)
- This affects many streaming services including YouTube app, Netflix app and Amazon Prime app.
- Since it's affecting multiple different 3rd party app, it's speculated that this is a bug on iOS rather than apps.
- It's reported that reboot or resetting network configuration will solve the problem, although only temporarily.
This is posted on Apple's developer community, and it seems like YouTube's team recognized this from how it seems on google's support thread. I don't exactly know how to contribute to solve this problem but I guess we have to wait or revert to pre-16.1 versions from backups to mitigate the issue for the time being.
If you need to watch YouTube video without least effort, you can use visit youtube from browser and it plays fine.