The Republic of Korea, T. R. Fehrenbach explains (in This Kind of War), had never seriously intended to halt at the old border:
It is very doubtful if Syngman Rhee, who lived to reunite his country, would have obeyed a U.N. order to stop short of the parallel, any more than Abraham Lincoln would have favored an order from foreigners to stop the Grand Army of the Republic on the Potomac after Gettysburg. Rhee issued orders to his field commanders, now serving under American command, to move north north no matter what the Americans did.
Whatever the ploy and counterploy of the great powers, it was in the vital interests of the Taehan Minkuk to expand to the Yalu.
On 1 October, MacArthur demanded the surrender of North Korea. Kim Il Sung made no reply.
At noon, 7 October, American units of the Eighth Army went across the parallel at Kaesong. ROK troops had already gone north days before.
[...]
There is every indication that, just as they had not expected that the United States would intervene in Korea in June, the North Koreans did not anticipate the U.N. offensive over the parallel. The shattered Inmun Gun had not been reconstituted after its retreat, and the extensively prepared positions along the border were not heavily defended.
It does seem strange a man so totally reliant on the USA for his very existence would act in such a manner seemingly heedless of consequences with his benefactors.
Rhee was not an American; he was a Korean. As such, he acted in the best interests of Korea, not the USA. And, the USA had clearly screwed up as a “Big Brother Ally”, in that they’d put all this money into South Korea, and then failed to actually leave them with an effective defense.
If it had just been South Korea against the North, without US aid? South Korea would have fallen, and stayed fallen. However, once we propped up the South and took them under our wing, we were beholden to keep them going or suffer the consequences elsewhere in the world with other allies. Moral hazards of doing what we did.
I’m still sort of ambivalent about Korea. They’ve always been a bit of an issue, there on the periphery of China, and not being Chinese. They’ve also been more than a little “off”, in that they’re not a particularly military culture, valuing their scholars and court more than their soldiers. It’s noteworthy that even when they were effectively resisting the Japanese back in the days of the Turtle Ships, the court in Seoul was doing all it could to undercut the admiral who was defending Korea. If a nation wants to survive plopped where Korea is, in between China and Japan, it had better be good at defending itself, putting a high priority on soldiers and defense matters. The Koreans, however…? Not so much, for much of their history. They’re kinda like that nerdy kid stuck in between the jocks and the greasers, constantly at the mercy of their more warlike peers.