Oh boy! I've been waiting for a question like this. There are layers of explanations for this, which might seem like the classic impulse of historians to complicate things, but really helps elucidate the nature of the historical moment.

First of all, from the perspective of Washington DC, the landscape of "Asia" at the end of WWII was a fraught and chaotic place. The familiarity of European imperialism was gone, shattered outright by Japan's wars in the Pacific and nationalist movements, eager to press the advantage and achieve independence. In Manchuria and Korea, SWNCC (State-War-Navy Coordinating Committee) members watched with anxiety as they saw Soviet Union armies stomping over the enfeebled Japanese control of Manchuria and Korea, compounding the challenges faced in the Chinese Civil War that were hard pressing America's ally Chiang Kai Shek (in the opinion of the Truman administration's cabinet, the least worst option available). The empires of Britain, France, and the Netherlands no longer had viable futures in Asia, although it would take years of bloody warfare against the Viet Minh to hammer that lesson home for France. FDR's trusteeship notion was one of many circulating on how to manage the future of a postcolonial Asia, although his death in office prevented that plan from coming about.

Compounding the matter? The US foreign policy establishment was distinctly Atlantic-facing. Dean Acheson, future Secretary of State and architect of the Marshal Plan, was an Anglophile, noticeable as much in his policy expertise as in his fashion choice. There was a coterie of China and Japan specialists in the State Department, but little knowledge outside of that. Korea, a former colony of Japan, became divided at the 38th parallel (to split the duties of US and Soviet occupational duties) in a late night session between two US officials who worked off an outdated map. Some OSS officers may have made good connections with would-be nationalists during operations in WWII, but their reports home had little impact on policy planning back in DC.

What would the future of Asia look like? I'm guessing nobody in Foggy Bottom in September 1945 would have figured that in five years, thousands of American GIs would be spilling their guts defending the regime of someone named Syngman Rhee in a place called Korea, while China's Nationalist government was clustered on the island of Taiwan, surviving by a single thread. Planning for the future of Asia during WWII was subject to rapid change and there were few people in Washington DC who knew enough to make accurate predictions.

In Europe, governments in exile returned to their original borders. France and Britain, the heart of America's alliances in Europe, were still France and Britain, wartime allies who had put aside their differences against a common foe. NATO, or the idea of a mutual, multilateral defense alliance, was really not too far off from what the Allied Powers experienced during WWII, albeit with the Soviet Bloc as the enemy this time. Of course negotiations were complicated and the future was fraught with difficulty, but from the American side, there was widespread and consistent consensus that securing a Western Europe friendly to American interests would rely on its economic recovery undergirded by a military alliance that would keep the wartime bonds alive where possible.

In Asia, there are brand new governments and nations to contend with, complicated by European powers jealously trying to hold on to their colonies, nationalist movements of suspect motives, and sheer instability. Would Syngman Rhee make for a good ally? Are we stuck with Chiang Kai Shek? (Bruce Cumings has found evidence that the US govt was planning to support a coup against Chiang in June 1950, when its attention was diverted by the Korean War). Who is this Ho Chi Minh guy anyways? Not only was planning more difficult, but also the legacy of the wars in the Pacific was /not/ wartime bonds, but animosity and hostility between new Asian nationalist governments.

Japan was the heart of America's strategic position in the Pacific and the target of the most planning during the war. At the end of the war, Acheson and his crew very quickly realized the importance of Japan's economic recovery and capitalist growth to the overall stability of a US order in the Pacific. But the legacy of Japanese imperialism poisoned relations between it and most other new nations, especially Korea and the Philippines. Korean and Japanese diplomats would not even speak with one another if they could help it in the early days, let alone hash out details of war reparations or the return of looted property. Only with the Korean War did there emerge the possibility of a multilateral alliance based around anti-Communism, initially titled the Pacific Pact. Yet Japan remained under US direct control and formally disallowed from having an offensive military and most other nationalist politicians, from Syngman Rhee to Pres. Quirino of the Philippines, always scored political points with their home audiences to flaunt their anti-Japanese credentials. The "allies" that the US saw in Asia were also highly suspect. Acheson and later John Dulles (rightfully) suspected that Chiang Kai Shek would use support from the US to launch ill-advised invasions of the mainland, dragging the US into another land war in Asia when it had barely ended the Korean conflict in a stalemate. Nobody really liked Rhee of Korea. Bao Dai, the French-supported emperor of Vietnam, was widely seen as hardly a sufficient national leader to attract away supporters from Ho Chi Minh. I could go on, but one gets the point. This is not an easy piece of real estate to build a multilateral alliance! Furthermore, a multilateral alliance inherently reduces the power of any one participant. Why would the US risk losing its leverage when it barely trusted its "allies" to try and take advantage of US support.

SEATO is the eventual Asian counterpart to NATO, cooked up by Dulles during the Eisenhower administration to contain Communist China. But the treaty signatories, which included Pakistan, parts of Southeast Asia, Australia, New Zealand and NOT Japan, all knew that the treaty was more formal than substantive. It was designed to give the US, or any other power, a way out of any real commitments to one another. In essence, it demonstrated common anti-Communist interests without constraining US freedom of action. The real levers of power were in the bilateral alliances, the "hub and spokes" system between DC and its respective allies, which aimed to maximize the dependence of Asian allies on the US, and the leased military bases which gave the US physical locations to project power out from without having too great of a marriage with its host country.

I think the point here is multilateral, NATO-like alliances are HARD to create. It's part of what makes NATO so remarkable that it has remained mostly intact for generations, even after the disappearance of its common enemy in 1989. The conditions that could make a multilateral alliance possible in Asia just wasn't there. Even 75 years after the end of WWII, the conditions still aren't quite there for the possibility of just a trilateral alliance between South Korea, Japan, and the US (no matter how much US diplomats struggle for it.) To understand why, one has to know what the world looked like at the end of the largest war the world had ever seen and how risky the future looked like in a moment fraught with contingency.

For further reading, I'd highly recommend Bruce Cumings' Dominion from Sea to Sea and his book The Korean War (excellent for a quick read), Dower's Embracing Defeat, Marilyn Young's The Vietnam Wars, Dean Acheson's memoir Present at the Creation, and Melvyn Leffler's Preponderance of Power. There have been few good studies of SEATO. John Franklin's dissertation "The Hollow Pact" is a decent overview. I could go on with sources and future directions for reading, but honestly 1945-1950 is one of the most written-about time periods in US diplomatic history, so there would be too much to list. Happy to answer more questions if interested!