There's no such thing as a torque sensing mechanism in this context, that would be a governor style that is more like a throttle on a gas engine, less governoring action, more speeding up with less load and slowing down with more load. You have a traditional diesel governor, so forget about that stuff.
The only speed control is the amount of fuel, so that button adds more fuel (possibly adjusts timing for better starting, but doesn't sound like it in this case). Yes, it will be tied into the governor linkage somehow, because that is the only output of the gov, more fuel or less fuel. Possibly no timing advance on this, ignore that I ever mentioned that.
So starting with the basics,
1) there is NO electric control of this injection pump? does it turn off with a key? or with a cable that you pull? or the throttle pushed to a certain position? And there are NO electric wires connected to the injection pump or anything connected to it?
2) fuel supply to this pump, put a tee into that supply line and a pressure/vacuum gauge where you can see it. Use the primer pump to see what pressure you can obtain. Then try to start it and see what the pressure does when it cranks, starts, and dies. The pressure should bleed down after it's died, but not at the same time it's dying.
3) take both injector lines off at the pump, and see if they are both moving an equal amount of fuel.
It would be unusual for many of these things to fail on a running engine without badly contaminated fuel, eliminate as much as possible before taking it apart. It looks like that governor housing is part of the front cover of the engine? I wouldn't want to take that off just to get a look. The injector pump elements that come off the top don't seem likely to show you much the way it sounds.
The info is out there if you want to look deeper at taking this apart, or see what's in there without taking this apart. There's a deutz german forum that google will translate for you, and many of the members would read english well enough to answer your questions if you can't find the answer.