Some more photos to clear any doubts about the mechanical aspects of the assembly.

 

With this part You only need an appropriately rated MOSFET,
and a Current-Limiting-Resistor for the LED.

Done.
.
.
.

 

It looks good, but I do not see any means of support for the circuit board except the solder connections to the motor attachment lugs. Both pressing the button and inserting/removing the connectors will transmit forces to those solder joints, as will any mechanical vibrations.

 

With this part You only need an appropriately rated MOSFET,
and a Current-Limiting-Resistor for the LED.

Done.
.
.
.

Thank you for your suggestion. I found some more info about that component here: https://www.bristolwatch.com/ele3/1g.htm

 

Just keep in mind that these devices are ( relatively ) SLOW.

They are not useful for PWM or High-Speed-Switching.

Their Outputs can be connected in parallel an/or in series to
provide either more Voltage or more Current to the Gate.
.
.
.

 

Note, thank you.