Sys 3 flipper circuit is pretty simple since Gottlieb never went full digital.
When you turn the machine on, even before starting a game, you should get around 60v at the flipper coils. This comes through the A15 sensor board. The chain is
48v supply > A15 board > anode side of coil
The flippers (as well as slings and pops) are enabled by the Q relay on the relay panel. This is normally controlled by Sol 31 (Q32).
If everything is working properly, and the game is running, when you press the flipper button, you complete the circuit to ground. The chain is:
48v supply > Q relay > fuse > cathode side of coil > flipper switch > ground bus.
The optocoupler on the sensor board monitors the power and triggers the two 2N3906s which are part of the switch matrix when the buttons are pressed. Since the board always gets power, the flipper switches are always active, which is why you can perform a switch test or step through rules on a Sys3 before starting a game.
If the coil is getting hot and the fuse is blowing when the game is running but not in attract mode, I'd go with the diode as well being the problem. Unless there's a wiring problem or a short somewhere in the circuit, or the coil is bad (which you say are all ok), there's not much else that could be causing the problem.
viperrwk