The flippers on this board are enabled by a relay, so the transistor you tested I'm guessing is just the one to enable to relay. Nothing you can do to it that I can think of will burn out a flipper coil. If it fails open then no flippers work, if if fails closed then the flipper relay is always on, but still needs the flipper button pressed to get the flipper to activate so it's not the problem.
The coil diode is a device to protect the power supply against freewheel reverse voltage spikes when the coil is de-energised, so it won't cause coil problems.
If the coil melted I'd look really carefully at the EOS adjustment. If it's not set properly then the full current through the coil will burn it out, this is most likely your problem I'd say. Much less likely, but possible, is insulation failure on the coil windings causing a short and then overheating.
