So it's not an 'internet hoax' that they're complicated. Source: I used to collect Leicas in the 70s and I talked to a LOT of repairers, and I have been inside a (broken) Contax myself, as described in my A History of the 35mm Still Camera, The Focal Press, 1984 (well pre-internet!).
First, what I call "Internet hoax" is the one that tells that only one person in the world can repair Contaxes the way they ought to.
Second, it depends on which Contax you speak of. The Contax IIa shutter isn't that complicated, it works with a basic system of cams driving the shutter shafts and the slow speeds escapement is a remote timer module that can be cleaned on its own.
Basically it's a Leica screw mount with a vertically running shutter (and a better viewfinder/rangefinder unit).
Problem is that the vertically running shutter relies on high friction because of the shorter travel of the shutter curtains and that the least delay in the first curtain journey departure makes the shutter cap and 1/1250 and 1/500 speeds don't work. Some corrosion in the sealed first curtain shaft can be a major problem because the shaft housing cannot be dissassembled for cleaning.
The Contax II shutter geartrain with its integrated slow speeds escapement is dramatically complicated, yes, but basically you never have to strip it down - a deep clean-up and typical relube at the shafts ends and at some other plances is enough because of how this very particular shutter works.
Most problems on the Contax II regard the shutter ribbons, the first curtain drum roller spring, and the shutter blades thin leather cords, but everything there can be replaced.
Source : I have myself been in five not working Contax II, one not working Contax III, four not working Contax IIa, and two not working Contax IIIa. All of them got back to normal work (and still do), including the III and IIIa lightmeter. And I'm just an amateur repairer, like Mike and some others out there.
I sometimes had to spend more than four hours on a camera but probably because I sometimes paid very much attention to each detail and often preferred to allow myself to rest between the service sessions.