The Stern circuit will pulse the coil even if the EOS switch is not working. It assumes the flipper is fully engaged soon after the button is pressed and starts pulsing to prevent the coil from burning in a holding situation. The purpose of the EOS in modern machines is to know when (if) the flipper bat is pushed back so it can apply full power to bring it back up.
MOSFETs do not generate meaningful heat when "off" and very minimal heat when "on" due to their low "on" resistance. When operated between on and off, significant heat can be generated. MOSFETs are best used as switches, either on or off. It is during the transition from "on to off" or "off to on" when some heat can be generated. The slower this transition is made, the more potential there is for heat to be generated. While the flipper is in hold state, the Pulse Width Modulation (PWM) function rapidly turns the MOSFET on and off many times per second to prevent the flipper coil from overheating. If the transition occurs "slowly", significant heat could be generated during an extended flipper hold. This is also why someone suggested removing the .01 capacitor that is on the MOSFET gate lead. Capacitors slow the change of voltage both on and off and could cause the MOSFET to be in the on-off or off-on transition slightly longer than without the cap. However, the cap is most likely there to reduce noise in the circuit or for the purpose Ed mentioned above.
I doubt the capacitor is the cause of the problem. If it was, they would have removed it from the design on future machines, which they have not.
I suspect the clamping diode associated with that MOSFET has gone bad or is weak. When you replace the MOSFET, put a new 1N4004 or better yet, a 1N4007 diode in the circuit. The diode numbers match the MOSFET numbers (i.e. Q14's diode is D14, etc).