I appreciate that you've taken a lot of steps to pin down the problem, but have you cleaned the contacts that the shoe slides on to? If they haven't been used in a while they might benefit from cleaning. If you have an analogue test meter set on the ohms range you could check the camera contacts to see if the meter needle kicks when the shutter is fired (a digital meter probably wouldn't be fast enough to react to the closing of the contacts).
If you draw a blank there, it's possible that it's a wire come adrift, dirty flash contacts or the possibly the component that acts on the contacts being out of adjustment. If I were to start looking, I'd begin underneath the top plate on the rewind crank side of the camera (left and right top plates are separate on the F2) purely because that requires the least amount of disassembly. If the problem is in there, it should be an easy and cheap fix - well as cheap as fixes are these days if you send it to a repairer. If it's deeper inside the camera and requires the lens mount/front panel removing, then of course it gets more expensive and you have to ask whether while it's stripped down a CLA would be cost effective at the same time. I'd like to think that a repairer would throw in a simple flash fix for nothing with a CLA (if indeed it is a simple fix) but somehow I doubt it.
Good luck and please let us know how you get on.
Best wishes,
Steve