Going in was long before the Yalu. The intent was simply to keep the south from being incorporated into the north. Americans' concern was not to be embarrassed by having to retreat from Pusan. There was no UN mandate to do more which is why it was a UN police action, not a US war. Anyway, America was war weary after WW II and had disarmed. Civilians could buy jeeps and M-1s. Optics for Norden gun sights were available in the first simple catalogues of Edmund Scientific. The focus was on Europe and we certainly didn't want war with the new Communist regime in China.
To add to /u/ApuleiusBooks' answer, before the war broke out, the U.S. was in the process of withdrawing its soldiers and heavy equipment from the peninsula in reciprocity for the Soviet draw-down in the North in (IIRC) 1947-8. If Kim il-Sung had twiddled his thumbs for another few months to a year, it's possible that no serious U.S. presence would have remained on the Korean peninsula at all. The U.S. already had expensive obligations to worry about in West Germany and Japan, and there was never as much domestic support for the Korean occupation as these. (Korea had been part of the Japanese empire during World War II and not a belligerent.) Moreover, part of the reason for the withdrawal was to deny Syngman Rhee (then president of South Korea) access to weapons and equipment that he wanted for a possible invasion of the North. The U.S. didn't want to antagonize the Soviets or the Chinese, and wanted to stick to the Korean reunification process that it was then pursuing through the U.N.
Invading the North to repel the invasion and deter Kim il-Sung from future attacks was one thing (that was the subject of the U.N. mandate that the U.S. had anyway), but occupying and rebuilding the country would have been something else altogether. Again, the Korean occupation was already a domestic political issue in the States and would not in any way have been improved by adding to it.
And finally, if the U.S. had forcibly reunified the peninsula under one government, a U.S.-occupied territory would then directly share a land border with China (hostile) and the U.S.S.R. (hostile). Nobody wanted that.