Just highlighting a few major flaws.
Once again, you think that the Wallies would not change a single bit about their strategy if such thing as the collapse of the Soviet Union was done.
Also, the Soviet collapse can mean a lot of things, including a surviving SU behnd the A-A line (the original plan), German units at Vladivostok, and also a huge number of fleeing Soviet troops which were not stationed in the Western frontier.
If you assume that the Germans could have defeated all the Soviet forces of 1941/1942 on their western front, as well as their units from all other sectors, they would suffer some extra casualties, no? The extra panzer group in AGS would not mean the bloodless defeat of all the Soviet military in Central Asia, Far East, Caucasus, etc.
Also, in order to redirect the extra units to your Barbarossa, as well as reshaping the production, you already sacrificed German contribution to the African Axis effort, the Ju-88 program and so on, effecting a "little" change in the situation in other theatres. For example, the Italians would likely to be ejected from Africa by late 1942, making all the North African logistical effort redirectable to the Middle East, not to mention Torch, which was not quite necessary or sensible in this timeline.
Also, you seem to ignore so many aspects of logistics, that I might call it the very nature of the thing.
First, the German supply system worked from depots, then the supply went to the "pipeline", meaning higher formations, and then the troops carried supplies, too. By the nature of the German war machine, and the operation you described in the Soviet Union, the German forces had to rest and refit first, build up resources on the frontiers; they did not have the supplies to immediately launch a grand offensive into Turkey and the Middle East. No forward depots were present around the Turkish border.
Second, if the locals (ie.: Turkish) are cooperating, it might ease a bit of the problem: guard duties, sabotage, local foodstuffs, local infrastructure, additional units, etc.: in absence the SU, the Turkish army could be equipped with relatively modern weaponry and increase their industrial output tremendously over time.
Third, again, even though I am not familiar with all the details (it's several lifetimes' study anyway), I know how the German and Wallied logistics worked. They produced matériel and trained soldiers at their home bases, shipped them to air/ports, put them on train and deployed them.
The LL did not mean the Persian Corridor alone. If the Soviets stop fighting (already described how many scenarios could that mean), and the LL does not work anymore, it makes no sense to quote people OTL about how unfeasible was to supply other fronts, on top of their OTL efforts.
What would make sense is a real analysis of the situation.
In your scenario, the Germans just finished a bloody campaign with extraordinary speed, which alone would cause a great drop in combat readiness. The airfields, railroads, bridges, supply depots, hospital capacities, etc. are all in bad condition near the Caucasus. There was also no prospect to take more resources from the locals.
On the Balkans direction, the sabotage is still a problem, the railways had low capacity, and because the Gibraltar was not taken, there was no real prospect to bring in more major ships to haul the cargo or help the invasion.
What we are dealing with is a single railway line with multiple bottlenecks and little ability for alternate routes. I already told you what was he situation of the Black Sea and Agean Sea maritime transport; on top of that, all losses of big merchantmen would be impossible to replace. Once the attack has commenced, and the Bosphorus strait is closed, no extra trasfer between the Black and Agean seas could be done.
The state of the airfields near the invasion zone were not in the best shape either. What we are talking about (improvement of the railway lines, airfields, forward supply depots, etc.) would take months at the very minimum to improve.
Meanwhile, the Wallies competed their preparations in August and December 1942, respectively. Their railway, which connected the Suez with Turkey's network, was improved to a standard gauge railway (20t/axle), with multiple entry points with metre gauge and also in connection with the Baghdad(-Basra) line. These two lines alone - without other entry points to port terminals of the Turkish railway system - would overwhelm the German logistics at least by 2 to 1 (the narrow gauge lines of the ME was designed to haul at least 150 tons per train, about the third of the standard gauge ones, and the Balkan load was 2/3 of the standard gauge load).
Not to mention that the very source you quoted claims 1500 tons / day between August 1942 and December 1942, and rising up from that point to 6,489 tons / day in 1944, with a peak month of 7,520 tons / day in the Persian corridor.
If the Allies upgrade the Basra-Baghadad railway, and the Germans remove the bottleneck from the Balkans railway, the ratio is still the same. 1 line against - at least - 2.
The Allies had a minimum of 1-1.5 years head start in railway development; and the Germans could not easily beat it, especially not with lack of sea control. But actually it was more than 1-1.5 years; both the Iranians and the Turkish invested an immense amount into their railway system. Something that the Bulgarians, Yugoslavians and Hungarians did not. The Germans had to make up that difference, too.
Then we arrive to the Achilles heel of the Germans' problems: the crossing of the sea. Even if they improve all the railways, etc. the next bottleneck is the sea transfer, which again, needs more than just ships and river barges. It doesn't matter how much supplies could be hauled to the Balkan peninsula; it would be a smaller amount that could be hauled over to Turkey. The Black sea ports could only help the effort if and when they are taken. Even then, it is a risky undertaking, for every loss is irreplaceable. What could help here is the redirected shipping from North Africa; but then again, the Wallies can redirect their shipping from North Africa, and I wonder if it is not a net loss in ratio for the Germans.
In order to support such an offensive, the Germans would need to face an aerial battle, too. Something that broke their back in Tunisia and Stalingrad OTL. The well-supplied and trained Wallies face the Germans close to the Turkish airspace; better deployment, superior intelligence, etc. would mean a huge attrition on the Luftwaffe. The oil of the Caucasus or extra production doesn't help in 1942; also, the Germans burned through 40% of their airforce in Tunisia in a very short time. Some months in the summer of 1943 resulted in a 15% loss of their force structure. Such attrition ate up all the increased production they had.
On the other hand, the Wallies had an ever-increasing capacity to supply large formations on the sites which the Germans have to attack, better trained aircrews, local support, better intelligence, etc.
“And while I am talking to you, mothers and fathers, I give you one more assurance. I have said this before, but I shall say it again, and again and again. Your boys are not going to be sent into any foreign wars." - FDR, October 1940