Most YouTube content requires the Flash plugin... try troubleshooting.
Open System Preferences > Flash Player then select the Storage tab. Click: Delete All
Now uninstall the Flash plugin then reinstall new > Troubleshoot Flash Player | Mac OS
Launch Safari From your Safari menu bar click Safari > Empty Cache
Now try a video.
BTW... if you have the ClickToFlash extension installed, that can prevent Flash based video from streaming. It can also be installed as a plugin in /Library/Internet-Plug-Ins.
And check to see if Safari is running in 32 bit mode. Right or control click the Safari icon in your Applications folder then click Get Info. If the box next to: Open in 32 bit mode is selected, deselect, quit then relaunch Safari.
My problem seems to have been resolved for now. Unfortunately, I'm not entirely sure which action I took fixed it.
The first thing I did was make sure I wasn't in the YouTube HTML5 trial. A month or so ago I suddenly couldn't play YouTube videos and figured out that I had been added to the trial. I opted out, problem solved.
However, when my recent problems with YouTube began I went back and checked to make sure I hadn't been put back in the trial. I hadn't, so I started looked for other solutions and posted my question.
I did the things you suggested (however I upgraded the Flash plugin without unstalling the existing one first). It didn't correct the problem. But on a whim, I went back once again to the HTML5 trial page and found I had been added back into the trial. I opted out again and the problems went away.
So, I don't know what to attribute it too because there was a short period of time where I wasn't in the trial and also hadn't done any of the things you suggested yet it still worked.
Anyway, I appreciate the help. Thank you.