Those two blue wires on the left switch look really close together. Might be perspective, but with how much insulation is stripped away there, that would bug me if it was in my game. I'd bend the solder end of the leafs away from each other or recut and solder them again so there's no so much exposed wire. No idea if that has anything to do with your issue here though.

Your problem is the EOS switch contacts. High resistance will drop voltage and make heat. High resistance will also cause higher current flow which can blow fuses.

The contacts need to be filed to clean them, they need to be adjusted in such a way as to apply more closing pressure on the contacts but still need to have an 1/8 gap in the hold position. One contact has a stiffener and should not be adjusted the other should be adjusted to add enough pressure to move the contacts a little bit toward the coil against the stiffener. The lane change switch gap is too small and needs to be adjusted. Also make sure the plastic piece of the lane change switch is not stuck to the EOS switch blade as it needs to slide on the blade. You may need to trim the plastic with a razor blade. And since you have the file out you might as well file the cabinet switch contacts to.

They come with an anti rust coating that needs to be removed with acetone before use. I can see gray burning on the contact as if there is a poor connection between the two contacts, remove this with a file. I have no picture for say but I can use your first pic as an example. Start by making sure the screws are tight. Then put the file in between the contacts and apply light pressure to close the contacts while filing them flat and clean. The single blade in your pic needs to be bent towards the coil more. You can see that the blade has an arc to it away from the coil, this causes the contacts not to be square and a poor electrical connection. High current thru poor or loose connections means heat. If you were to remove this switch you would see how little contact they are making. Now besides a square contact there also needs to be a certain amount of pressure holding the contacts together, if this is not enough the flipper will get weaker on a long game. The contact pressure is not preadjusted and needs to be set any time you work your flipper mech. Once you set the pressure then you can check the EOS gap by only pushing on the coil plunger. If you do it any other way your gap will be off by the amount of play in the flipper parts. All lot of the time you need to do this 3 or 4 times to get it correct. As you change the gap it might change the contact area or the contact pressure. Once this switch is done you can do the other stacked switches.

I noticed in the photo that the arm on the flipper pawl that pushes the EOS has a worn portion on the rubber. Not sure if this could cause a short, would pay to replace it in any case.

The rubber has been sticking to the hot switch, you can fix this with a piece of heat shrink.

Ok, tested and with the clip in place the flipper works.

Removed the alligator clip and the flipper is still working. Obviously the eos switch was slightly out of adjustment but looked perfectly fine visually. Weird.

Sorry, wrong thread lol