Adobe aren't supporting their flash player on amd64 right now. The cognitive dissonance gets a little draining, anyway, and I've seen the hoops I'd have to jump through to get the 32-bit version running. So I'm going to try tracking gnash trunk for a while.
First impressions: gnash seems easier to build than it used to be (or maybe I just read the instructions this time). I chose the AGG graphics backend, with gstreamer and gtk. I also chose to install the browser plugins system-wide. The bzr repository ships copies of the necessary Mozilla libraries, which I'd usually frown upon as a packager, but it did mean I didn't have to worry about them.
So the whole process looked like:
bzr branch http://bzr.savannah.gnu.org/r/gnash/trunk
cd trunk
./autogen.sh
./configure --enable-renderer=agg \
--enable-media=GST \
--enable-gui=gtk \
--with-plugins-install=system
make
sudo make install
sudo make install-plugins
The README file was very helpful, as was the configure script output.
So, off to youtube... "An error has occurred, please try again later." Hmpf. Off to read the mailing list... there was a problem discussed a while ago about cookies that YouTube has started sending. I checked the patch discussed at the time, and a version of it seems to be in trunk, but the symptoms are still happening for me. But... disable all cookies in the browser, and it starts working. (There's a chance that libcurl might not be configured properly.) I'll have to report the problem or track it down further, but for now, I'm sitting back and declaring success.
Co-incidentally, the video was discussing another cookie problem. ("Put pinky down. Down pinky. Good.")