When you turn the machine off, the Lock Relay de-energizes, and then back on, the Game Over Relay
energizes. So when you say you try again, you must mean that you fire up another game which the Start
Relay will energize the Score Motor. I'm sure you tightened the switch stack screws on the Score Motor's cam.
Have you checked the Score Motor's run-out switch making sure it has good contact through it's entire revolution?
I know, once working properly, it remains working properly, but at the start, the Score Motor's not tuning at all,
or at times it starts (because it sometimes scores), but it doesn't complete it's revolution and change players/frame..

I believe the coil you're referring to on the coin mech is the lock out coil. If you drop a coin when the machine
is off, that coil not being energized will change the coarse of the coin so it travels to the coin return. It wouldn't
have anything to do with the issues you're having.
I forgot, but did you check the Slam/Anti Cheat switches? If the machine is abused (kicked, punched, picked
up and dropped) the one blade with a small weight at the end will break away from its mating switch which will
break the circuit to the Lock Relay ending the game. If the set/sets of Slam Switches aren't making good
contact, it could cause the Lock Relay to more easily de-energize.
A Slam switch could be on the Coin door, under the coffin, and/but almost always there's a set on the side inside
the head. I had a few Venuses and still have one, but I can't remember how many Slam switches there are.
I'm thinking the one inside the head may be the only one..

Yes true, but if it has just a slight amount of tension, a little vibration maybe
could cause the problem, but like you say, after a couple of start ups, it seems okay.
The next time you start it up cold and this issue happens, maybe check if the
Lock Relay de-energized..