Pictures are better than youtube. Everytime I pause the video the resolution is bad. It is fixable and none of the parts will cost much.

But anyways it looks like a series resistor between the FET and the chip burnt up too. If that resistor is burnt I think that is bad news for the U3 IC chip.

Do you have a logic probe? Pull the fuse. Put the game in solenoid test. Probe Q6 of U3 which is at P16 while the game is in solenoid test. You want to see mostly low with a blip high during the period the transistor would be on during test. You can cut away the remains of C33 for this test.

I think you will need to test / replace.

U3 - 74HC273
R83 - 2.2K
C33 - 10nF
D22 - 1N4007 (this may be on the circuit board or on the coil, not sure)
Q22 - 12N10L or better (15N10L, 22N10L, IRL540)

The coil connected to Q22 also needs to be checked. Not shorted. Lugs not shorting to anything nearby. If the diode is on the coil, it can't be cracked, broken or missing.

If this is out of your wheelhouse I would send off the board to be repaired. Will be easy work for someone that repairs these boards often.