Yes, it does require a bit knowledge on electricity. So if you're not sure, leave it or get someone else to do it.
A replacement thermostat would be around $30-$40 if you did it yourself and sourced the part. But to have a fridgy look at it you could be paying up to $200 I'd guess.
Sounds like you could bypass it completely and allow the external thermostat to control it.
My last two acquisitions were given to me as they were destined for the tip, both about 10yo. An upright frost free freezer, good as gold with a new $30 thermostat. And my 520L upside down frost-free 6 keg fridge, good as gold with a $10 part!
Your quite correct, simply pop out the old thermostat and see if it has 2 wires and if it dose then unplug them and plug/rewire them into the cool contacts of your STC-1000 then give your stc some power on the power lugs only (dont wire across to the cool or heat contacts like normal).
That will only work if the thermostat is gone though, if the compressor is gone you have a new esky! :lol:
So unplug it from the wall and tell us how many wires your thermostat has?