When R51 is burned up like that it is a pretty much guarantee that all three of the high voltage transistors will be shorted which makes HV input voltage = same as ouput voltage (~230v) and the trim pot will have no effect on HV output voltage control. A V-in to V-out shorted high voltage should not blow the output fuse on the driver board tho so its possible you have a bad display or some other kind of short on the hv output voltage.

I would repair the high voltage circuit... Replace R51 and all three transistors.

Unplug all the displays. Power up and check the HV output. If its now looks correct at about 180vdc. Plug in one display, power up see if things are working. Power down. plug in another display, power on .... until you have all displays working. If you find one display that blows the fuse repair or replace it.

in your picture of the back of the board the transistors are the holes labeled E B and C for emitter collector and base. The collector is the case of the big metal can transistor and is electrically connected with the heat sink mounting screw.

Probably a good idea to replace the caps but on these driver boards they honestly are fine more often than bad. In this case I doubt the capacitors have anything to do with problems the OP is dealing with. At times I think the rush to replace electrolytics is overblown. Everyone likes a good cap kit job tho.

With high voltage and conductive crud you may be able to blow the fuse. Check resistance of the high voltage rail to ground and compare working to none working, see if the +185v is shorted or low resistance to ground. Check VR1 to see if it is shorted.

You can wash the PCB with soapy water and a tooth brush. Then sit in front of a fan to completely dry. Alcohol or naphtha will displace the water and make it dry quicker. Maybe avoid soaking the glass. They are sealed but solvent or water might contribute to the seal breaking and gassing out.

Power off display removed from the game. DMM on resistance measure between the HV connector pin and the ground connector pin. Compare to working display.

VR1 check. DMM set to diode check. See if VR1 reads short, 0 on dmm. Compare to working one if you are not sure.