'70 fuel gauge-weird problem

An old switch is an excellent suspect.
Keep us posted.
I am not familiar with this conversion so I assumed digital readouts.
Either way I am sure “solid state” involves transistors, diodes and other things.
All I can figure is that if all your gages are feed off the same circuit there must be something different about your fuel gage which stops working when the engine is started.

What seems logical to me is that solid-state devices sometimes require a certain voltage to operate properly (gating things one and off)
And a voltage drop to start the engine would cause a voltage drop.
It is hard enough to trouble shoot when one knows what one is doing and actually can see something.
A bad switch certainly is as reasonable as anything else.
When I read the last part of your post, I understood that the gage worked after a set pattern of bumping your starter, returning to the “run” position and then starting normally without any further switch movement.

I had been playing around with the ignition switch a bit, what I was doing is turning it forward enough to where you get a fuel gauge reading, hold there a moment, then quickly engage the starter and back off "ever so slightly" without starting up. At this point, the gauge still shows a fuel level. Then I go ahead and start the car, and the sunnofavitch then seems to work properly. Of course, who the hell wants to go through this all the time????
There are no additional lighting or gauge issues