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I'm not sure there's any strong or direct connection in the academic sense. My recollection from reading the rest of the Feynman story that was quoted is that the wobbling plates problem lifted him out of a self-perceived slump or depression. So what's important is the impact that working on a fun and unimportant problem had on his state of mind and emotions, more than any particular insight or inspiration that problem had on QED stuff.

It's not that dynamics of plates gave him insights about QED. Working on a simple problem have helped him to get back into working rhythm of his phd years.

I don’t have full explanation of the connection that would be satisfying. However, I think it mainly comes down to symmetry group of the spinors and their 720 degrees symmetry. It is called plate trick for a reason after all: https://en.m.wikipedia.org/wiki/Plate_trick