Well, there are complex issues at work, and some partial truths. Intervening in Iran was always a complicated matter due to the existing agreement (signed in the interwar period!) between Theran and Moscow, that grants the Soviet Union the right to intervene in Iran if the country is used by a third party to stage attacks on soviet soil... So any intervention in Iran, larger than Eagle Claw (and that was himself quite large for a SOF operation with the seizure of two airbases...) could have landed the US in a difficult position. Add to that the fact that Carter was taken by surprise by the revolution (but again he was considering serious retaliatory measures after Claw's fiasco, but due to the Walker Spy Ring, the Soviets got into alert with their southern military districts...Certainly, oil supplies from the Persian Gulf were important, but do bear in mind that the US didn't defend the regime in Iran against hostile takeover, either. In many ways, the loss of Iran represented a much greater strategic setback that Kuwait would have.
Priorities change, but there are definite limits: the British didn't deplete BAOR to send troops to the Falklands; nor did the US empty out their bases in Germany to fight in Vietnam.
Said that Carter also created the RDF, then CENTCOM for such eventualities. Yet one of the big issue with the RDF/CENTCOM, something that H. Norman Schwarzkopf touches in his book, and you can see directly in Mark Herman's Gulf Strike early scenarios, is the relative speed (or lack of thereof) of deployment. You can send one brigade of the 82nd, one MEU, the Rangers, quite quickly, then you have a significant buildup. Second lift will be the rest of the 82nd, a MEB coming from Diego Garcia and airlifted, and the then the 101st. The latter requires a whole wallop of air transport capacity. The big punch would be the 24th Mech, and one of the first two Marine Divisions. And these would come in the third lift. So 'fast' option are quite limited, and you also require local support. Something that was not that easy to get on the fly in 1979. In the T2K world even of 4E, well, I assume the 24th has not been reflagged, and it keeps one brigade in Kuwait. The whole XVIII Corps is available (as it was in the 1st and 2nd Editions) to be deployed there. Certainly I see them going there rather than in Europe. Always sent them there in the TWW combined game, with one USMC division to bother.
The big thing is... SL-7 fast transport. They are a precious commodity, do you send them to move the 24th to the Gulf, or another one or two heavy divisions to Europe? Considering the key role of oil in modern warfare, I always sent them to the Gulf.
As for the two other claims. Operation Corporate troop deployment were constricted by logistics. There was just a fixed amount of amphibious lift and sustainment available. When Julian took the 3rd Commando Brigade down south, the bulk of RN amphibious assets were committed, including 'Ships Taken Up From Trade'. When General Jeremy Moore came down with the 5th Brigade he had to use civilian shipping. On top of that, Sending mechanized and armoured regiments down south from BAOR did made no sense at all, considering the absence of the road net, and Julian 'mistrust' )at the time) of heavy armor. Well, I found difficult too to see where anything bigger than a Scorpion could have been employed...
In Vietnam... well the USAF did indeed transfer units straight from USAFE to 7th Air Force (Operation Freedom Train) . The army also ran down units in Europe of NCO and volunteers to keep MACV full strength. That in turn created the so called 'Hollow Army' period.
Said that, all major NATO partners (US, UK, and France) had a reserve of expeditionary units to be used away from Europe. Remember that horizontal escalation (or rather the ability to react to) was a principle of US conventional doctrine from Kennedy onward.