It seems to me this is almost certainly an issue on YouTube's end, not with OS X or Safari, given a few things that have been mentioned:
1 - The issue started today, not yesterday. If the issue was introduced in the new Safari, it would have started yesterday, when it was released, not today.
2 - The issue is also coming up for people who didn't update to 10.1 yet - people still using 10.0 are also getting the issue. So it definitely isn't caused by the update from yesterday.
3 - Changing the browser agent fixes the problem. All that does is change YouTube's server to treat your browser as if it were a different browser. This doesn't actually change your browser.
So all signs point to YouTube having an issue with processing the agent string of Safari 8. Perhaps they maintain a whitelist or blacklist for which browsers should be able to access YouTube videos, and they accidentally removed Safari 8 from the whitelist (or added to the blacklist)... or perhaps they did it on purpose, to spite Apple and Apple's users. It wouldn't be the first time Google attacked Safari - remember when they exploited bugs in Safari's cookie policy so that they could track Safari user's via their ad network even when users explicitly asked not to be tracked?