You are quite right, tactically speaking.
Given that the US and UK were engaged in war with Germany, if they broke through in the Gap they would be more likely to drive into Austria, attacking their main enemy, rather than divert east to protect a (sorry) quite unimportant country.
Serbia is massively important in your eyes, understandably. Hard to see why it should have had similar importance in the eyes of western strategists in 1944/5.
Serbia, as it turned out, didn’t even play a particularly important part in the Cold War.
Serbia, as it turned out, didn’t even play a particularly important part in the Cold War. Serbia/Yugoslavia's role in the Cold War was not completely insignificant.
- Early in the Cold War, when it was part of the Bloc, Yugoslavia posed a threat to the Allies in the Mediterranean. Right after WWII, Yugoslavia briefly threatened hostilities over the disputed city of Trieste, and there were even shooting incidents. In 1947, Rep. Richard Nixon, on a junket to Europe, witnessed Communists rioting in the streets of Trieste, which influenced his view of the Communist threat.
- Later, Tito's falling out with Stalin helped bring about the defeat of the Communists in the Greek Civil War, a victory for the West in the Cold War.
- The Kennedy administration tried to woo Yugoslavia even further away from the Soviets by giving it military aid, a move that stimulated protests in the US which contributed to the conservative movement that was beginning to gather steam.
- Yugoslavia had a hand in the 1973 Yom Kippur War when it shipped supplies to the Arabs. There were reports that Soviet air and even ground troops had been moved to Yugoslavia, ready to intervene in the war.
Yugoslavia may not have been one of the major players in the Cold War, but it did play a role.