Safari definitely supports HTML 5 on YouTube without Flash.
You may have a Safari incompatible extension or plug-in that's prompting the, "This video is unavailable" ..
From the Safari menu bar click Safari > Preferences then select the Extensions tab. Turn that OFF, quit and relaunch Safari to test.
If that helped, turn one extension on then quit and relaunch Safari to test until you find the incompatible extension then click uninstall.
If it's not an extensions issue, try troubleshooting third party plug-ins.
Back to Safari > Preferences. This time select the Security tab. Deselect: Allow plug-ins. Quit and relaunch Safari to test.
If that made a difference, instructions for troubleshooting plugins here.