Several allied evaluations noted the relative short service life of the tracks and drive sprockets of German tanks. One soviet test shows the teeth on the drive sprocket of the king Tiger were wore down almost completely after a road march of less than 100 Km. What is the fundamental flaw in German sprocket or track design that led to this?

From what I read and from what some other fellas have posted here, it is because of the slack in the track makes everything wear faster.

T-34 has slack track too, but the tracks seem to last much longer.

Take a look at the drive cog on a motorcycle with a loose chain, the teeth end up looking like waves. I suspect that the weight of the Kingtiger and so kinetic forces between the teeth and track answers a large portion of this question. Another aspect could be that as the war went on the qualities of metals dropped.

Both the (King)Tiger and the Panther were seriously "overweight" when build compared to the initial design. Panther was planned around 30to and ended up at 45 as an example, Köttel was even worse.

But because both the WaffenSS and Adolf needed something with a long thick barrel to compensate the tanks went into production with too weak a drive gear (Thankfully in case of the Panther(1)) over the objections of the engineers and parts of the tank force. That plus drops in quality caused the problems

Other vehicles used the same track design (IIRC ALL german build tracked vehicles used it) but in a more balanced style.

(1) The engineers had a working solution that (together with the HL230 engine) WOULD have allowed the Panther to become the new standard plattform replacing PIII, PIV and variants in production (It was easier/faster to build as well). That would have given the germans a fast, well armed/armored plattform in the Leopard I weight range...