I think I might have cracked it. Nothing to do with being signed out, or Ghostery, or caches, but a setting I have not had cause to look at before: Safari > Settings for this website > Enable content blockers. This seems to be ticked by default. Unticked it for Youtube and so far this seems to have done the trick.
I have no idea if there are any downsides to taking this step, but it worked even with Ghostery active on Youtube so presumably unwanted stuff will still be blocked.