Yes, there was, as of February 1943, the Inspekteur der Panzertruppen was Heinz Guderian. However, it was a post in a large, monolithic and bureaucratic, and heavily regimented institution. It did not function like a bunch of ten-year olds trading baseball cards.
The reason they were distributed like that is because they were part of organizations, the Panzer regiments and battalions that were part of Panzer and the new Panzergrenadier divisions. They had been organized according to the doctrine developed between c. 1929-1939, which utilized two types of tanks, the Panzer III and the Panzer IV, in its tactics. The desired organizational ratio chosen was three companies of Panzer III to one company of supporting Panzer IV, while the ratio achieved in most cases was two-to-one.For example, there were 49 Mk. III with long 40mm with HG in Sicily, and 432 on the Kursk front. A little wheeling & dealing could have seen a trade of several of the medium tanks for each Tiger sent east, a good trade for all concerned.
Deployment of the 5cm K and 5cm L armed Panzer III in 1941 and 1942 did not change that distribution, nor initially did the deployment of the 7.5cm KwK L43 and L48; they were typically distributed on a one-for-one basis for the earlier model Panzer III and IV in the organizations. It was after Stalingrad and Tunisia, when the wave of reconstituted Panzer organizations were created to replace those lost that the situation changed. By then, it was decided that production should focus on the Panzer IV with the long 7.5cm guns, while production of the Panzer III switched over to the new Panther, which would replace both. So, as reconstituted, units like 14. and 16. Panzer Division dropped the Panzer III, while others, such as 26. Panzer and HG, kept a mix of Panzer III and IV. It was further complicated by the deployment of the Panzer III (Flamm) to some of those units.
The Tiger battalions were a slightly different matter, since they were non-divisional units, intended as an augmentation for the Panzer Korps...thus one per SS-Panzer Korps as they were organized, and for the renamed Armee-Korps (mot), which became III, XIV, XXIV, XXXIX, XXXX, XXXXI, XXXXVI, XXXXVII, XXXXVIII, LVI, and LVII Panzer-Korps, March-July 1942. Thus, 11 schwere Panzer Abteilungen (Tiger) were originally intended.
Of course, that plan didn't survive, mostly because they did do what you suggest, by shuffling some of the early Tiger organizations (elements of 501. and 504.) to Africa, so they could over-match the Allied tanks there. It didn't work, but the impulsive deployment left the remnants of those elements in Sicily, then Italy.
Um, ZITADELLE was a German offensive, so Stalin could hardly have timed it to match any move in the west.And yes, Stalin was a bastard, and not nearly as competant as some people seem to think. His big victories always seemed to be dependant on being timed to match the big moves made in the west; Stalingrad with Torch, Kursk with Sicily, and Bagration with D-Day.