1950 was not 2013
I think US would go for South Korean even if they were alone in it.
Given what terrible shape the US forces of the day were in - drastic force reductions, discipline and training changed to produce a 'civilised' army, Japan garrison at half strength with worn or nearly worn out equipment - the US was in no position to go in alone. Task Force Smith's fate was not an accident.
The account in 'This Kind of War' by T. R. Fehrenbach, originally subtitled "A Study in Unpreparedness." is both illuminating and horrifying on this point. The chapter 'Proud Legions' was, as I understand it required reading for any US officer selected for promotion to General.
The original question of whether North Korea could have won really requires a debate about what stopped them from taking Pusan? Were the Northerners preordained to be too weak and poorly supplied by the time they got there to actually be able to finish the job? How much difference would better preparation on their have made?
If Pusan had fallen would the Inchon invasion still have happened? Think about the political chaos in Washington with each party blaming the other both blaming Truman, the press making it out like the end of the world and MacArthur putting in his own two cents worth from the sidelines. Would there have been the political will to continue the war?
If the Soviets had been present to prevent the UN joining how big a difference would that have made in the first few months? I am still thinking about the survival or otherwise of the forces trapped in the Pusan pocket.
Would the US have stripped the Japanese garrison and then started systematically stripping forces from around the world to go to Korea if their allies had not thrown in with them? How much damage would the United States allies NOT coming to the party have done to the basic idea of the US being obliged to protect those same allies? The political leaders of the era had grown up with isolationism, that was their normal world. Keeping US forces in Europe had involved a lot of emotional debate in the late 40's, at least for the Republican's. The debates might have flared up again with a different outcome if Britain, France, Belgium in particular had not put men with rifles on the ground.