Looking at the schematic for the RevD board (which i assume is similar to your RevA) we can see that the circuit for Q1 through Q9 are similar, therefore we can expect your meter test readings to be exactly the same on each. Since Q7 and Q8 are not, we can assume the driver transistor and/or the suppression diode on those circuits is bad.

Driver MOSFET - https://www.mouser.com/ProductDetail/Nexperia/PSMN027-100PS127?qs=5DYmZrWuUEqupow7jHI3kg%3D%3D
Diode - https://www.mouser.com/ProductDetail/Micro-Commercial-Components-MCC/SK510A-LTP/?qs=%2Fha2pyFadug4ucAn58i43aqQjoM8bBvCQz5%252Bh5smqP1hT0Elza8mug%3D%3D

The MOSFET is easy enough to replace (through hole) but the diodes will be a bit more tricky as they are SMD. Shouldn't be too tough if you have a fine point soldering iron since it is only 2 leads (not as hard as replacing a SMD IC).

The GI driver transistors look good to me from your readings. However of you want to order some just in case they are here:

https://www.mouser.com/ProductDetail/ON-Semiconductor-Fairchild/FQP20N06L/?qs=%2Fha2pyFadujhaj0xBReI9xUAnlzFOBeKo69xT23Kvbk%3D

Can you CAREFULLY look at D9 (the one with the zip tie through it at the top of your picture) and see if either lead is broken off? These would vibrate until one of the legs would crack, causing an overvoltage to the 6v supply (this includes GI).

If that is broken, you should get a replacement for it as well:

https://www.mouser.com/ProductDetail/Vishay-General-Semiconductor/MBR760-E3-45/?qs=%2Fha2pyFaduiIANWfweiZVnqRONx8UmAa7nSQK1vptCZzj%252B8lEkARlQ%3D%3D

Look at the leads not at the board but as they enter the black body of the Diode. Push on it side to side to see if one of the legs is broken off.

Hmmm, OK then I really have no idea what caused the meltdown. It could have been any number of things.

Maybe the suppression diode failed which caused the driver MOSFET to fail which failed shorted, which caused 2 things: the coil to melt and the voltage be pulled down on the rest of the board causing other functions to fail like the GI.

Really impossible to tell for sure without repairing the board.

You could test to see if the rest of the node board works by mounting it in your game and connecting ONLY CN1 (the ethernet looking cable) and CN6 (the main power input). Do not connect the solenoid outputs. See if the game boots without the Node 9 not found error.

Technically you should be able to connect everything BUT CN7 and CN8 (those are the solenoid output connectors with the shorted MOSFETS).

The game will not be playable (those solenoids outputs are critical) but you can see if the node board works. This should give you a good indication if the board is easily repairable.

Also, when you get the new node board, make sure to set the switches on the new board, just like on the old board. This sets the board to respond to the Node 9 address. Without it, the game will not know where to "find" the new board.

I was thinking the same thing about fuses. I just might add them as a preventative.

I'd expect a coil that looks as burned as that one to be shorted internally. You do not know if this occurred first and blew the node board transistor or if the node transistor went first, overheated the coil which resulted in the coil shorting.

Yes, there are 5v and 3.3v regulators on the board that make their power from the 6V supply. Seems the damage to your board is more extensive than just the driver transistors. U7 and U8 are the regulators.