Who ever plugs them in externally? Sounds like a silly thing to do
Yes indeed you can pull the existing thermostat and put the new controller in right where the old one used to be. There's normally a lot of room down in the compressor compartment for doing exactly that. You can either mount it so the cover plate comes off to adjust the controller (easy) or so the controller sticks out through the plate (takes some cutting of the plate). I don't fiddle the controller much, so the pull the plate to adjust the controller approach works fine for me.
One thing to check - the sensor lead on your eBay controller may or may not be long enough to get up into the freezer compartment. It's worth checking before you decide on what you are doing.