I think the shutter mechanism is quite similar in both cameras, although the flash sync differences are reflected in many parts of the camera. The basic design flaw with Contaxes is that most of the bits that need to turn when winding the shutter also need to turn when firing it, unlike the Leica (and almost everybody else) where such gears are uncoupled when firing the shutter. Having the shutter cap at high speeds is a very common problem when things get crudded up a bit, and it is quite a chore to get at these parts. But if your shutter is not firing at all, I think you may have something more basic, accessible by just taking the top off, an easy job.
It may be that the slow speed escapement is jamming. They need to be squeaky clean to work reliably.With the top off, remove and clean the slow speed assembly in mineral spirits or just about any other sort of volatile solvent.
With the slow speed escapement removed, you will see a maze of levers, pivots, shafts and springs. The operation of all these bits is far from obvious, and I have come to the conclusion that at least some of these moving parts are there just to move out of the way of other moving parts. Of the two rotating shaft ends, the right hand one is the leading curtain and the left hand one the trailing curtain. Don't adjust anything here unless you are sure you know what you are doing.
Holding the camera upside down over a bowl, brush the exposed gearing, levers, etc over and over with mild solvent such as varsol. You will probably see the drippings get brown and sometimes show solid flakes. Don't overdo the amount of solvent, and take care to keep it off the rangefinder glass surfaces, as it can quickly spoil the clarity and contrast, and they can be difficult to clean. This can often free up a balky lever and cure shutter problems. Try to keep the camera inverted until everything dries out. If the issue is deeper inside the gear trains, it is a difficult and expensive job, which can cost more than the value of the camera.
Cheers,
Dez