No, it wouldn't. ContentID is already used by Google for that exact purpose. They appear to be fully in favor of enforcing IP law provided the owning party raises a complaint.
If SynthID works, this would allow people to tag their own videos with a watermark that is invariant across various levels of compression and editing. It would enable automated scanning of YouTube videos for uploads and the consequent class-action lawsuits.
Because of that, I can fairly confidently say that this doesn't work. However, it will function to divert some attention for a while. And that's what Google and OpenAI intended in the first place.
Class action? Wouldn't you first have to attempt to follow the official process, filing a DMCA request? And I assume that Google would likely respect that since (unless I'm uninformed) it seems like they typically have so far? I really don't see how this escalates in a way that leaves Google at fault.