Is there a clever
tool that releases the sprung hinges, or some other technique that I
am missing?

Is there a little fold-up 'hinge' that catches on a little spur on the
bottom of the 'arm' of the hinge? It's possible that if there is that it
might not quite reach the spur if the internal rollers inside the oven are
worn, so it looks like it doesn't do anything useful.

Al.