"Manual degausser" functions in original arcade cabs are mostly very simple. It consists of a button (appropriately AC voltage rated) with a spring inside, so that the circuit is only closed while the button remains pressed. Nothing more complicated than that.
Note that degaussing coils suck a lot of power and get hot fast, so should be operated for no more than a few (2-3) seconds, and not operated repeatedly over a short time (let it cool down). If you want to run with a manual degauss function, probably also a good idea to think about putting a fuse (and a line filter) on the AC input to your cab.
For TVs that degauss automatically on power-up, they typically switch off power to the coil using a thermistor (or some other temperature-controlled method).