There's nothing like applying the scientific method when you need to solve a problem, and that is what you are doing. Personally I would not have expected the double diaphragm setup to work, because the doubled flap valves might be too thick to work, but I haven't tried it, so that is just prejudice, not science. You've found that it didn't work, but not why. If you now try the mylar one alone, and it works, the only possible reason will be something that is different between the double diaphragm and the mylar diaphragm alone.
If the interior of the carburetor was dry, no fuel was being pumped, so the problem was most likely in the fuel pump. You put the old diaphragm back in and it worked for a while, which further supports the idea that some aspect of the double fuel pump diaphragm was the specific cause.
Because the baggy old diaphragm had been working for years, it will quite likely continue to work for a while longer. Hence I do not think it would have been the sole cause of the machine flaming out and not restarting. However if the output from the fuel pump was pretty marginal with the old diaphragm, combining that with low fuel level and a lower starting throttle set-point might just keep the machine from starting. Filling the fuel tank and putting the throttle cable back the way it was originally might get it working again, for a while and in a fashion.
If you continue to go step by step and apply careful reasoning, you should get to the bottom of this. Personally I'd try the mylar pump diaphragm as the next experiment, rather than try to get some action out of the baggy diaphragm. You've said that is what you plan to do, too. I'll look forward to hearing how it goes.
