They currently allow Brave which does that.

If a major website starting blocking user agents for any reason other than compatibility then all that would do is trigger (finally) the deprecation of user agents strings as browsers would all just pretend to be each other.

User agents are trivially fungible thing in this. That's the nature of the internet and how connections are established. Personal hardware constructs whatever packet that is going out and there's nothing special if that packet comes from a sanctioned or unsanctioned client. There's no way for a provider to distinguish the difference between what created an acceptable packet.

It's difficult for me to envision how that's rigorously enforceable to prevent something outside of the app (like a browser) to access something ultimately consumed by an API without resorting to (expensive and crackable) digital rights management at the hardware level. You can run down whatever train of thought you want, but ultimately since the client is running on machines controlled by the consumer, it can be cracked, reverse engineered, decapped, or whatever. This has been the end state for every attempt the industry has made to control user behavior. DVD CSS, Bluray, Sony's CD rootkit nonsense, every game console, satellite TV cards, analog cable TV, on and on, all broken.

I'm confident google/youtube wishes there weren't applications like FreeTube and yt-dlp. The fact of the matter is it's practically impossible to exclude them but still keep their own ecosystem functioning at an acceptable cost and with consumer buy in.

Because they don't want to get sued for monopolistic practices, I guess. Since the Goo owns Chrome and Youtube, Youtube blocking non-chrome user agents would be actionable.

Even if they whitelisted Safari, that would still be actionable.