The problem is nothing to do with the EOS switch, you have an open circuit on the "hold" winding of the coil.
The EOS switch normally shorts the hold winding allowing maximum current through the thrust winding.
When the flipper reaches the end of stroke, the EOS switch opens adding the hold winding to the coil circuit to reduce current.
Because your hold winding is open circuit, the coil is collapsing causing the flipper to retract until the EOS switch makes contact again. This is then causing the machine gunning.
Carefully inspect the two very thin winding wires on the coil going to the middle lug and the lug with the two dark brown wires soldered, the thin wire might be broken at one of the lug.