Why is the Korean War often neglected in Western popular culture?

I would argue that the forgotten, nigh-amnesiac aspects of the Korean War is precisely what makes it so interesting for me and other scholars who still center Korea in broader surveys of the American military, the Cold War, and postcolonial violence. Rightfully, as you point out, the three year conflict on the Korean peninsula has massive importance to the present day. But not only is it relevant to existing tensions between North and South Korea, it was transformative for the entire world at the time and it is relevant for any discussion about modern Asian history.

But back to the original question. What about the Korean War made it so slippery? It's a good question. WWII was transformative for American conceptions of the self and long after the last living American veteran has passed away, stories and cultural media about WWII will likely persist. Vietnam was also earth shattering in its cultural impact. However, I would argue that Vietnam has faded in its cultural impact in the US in the decades since, at least compared to the sheer amount of movies and songs that came about in reference to Vietnam from the 1960s to the 1990s. There's a very good hypothesis for that, which is relevant to your question about Korea.

In the US, Korea was a confusing, forgotten war even while it was happening**.**

"If any war that our country ever engaged in could have been called a forgotten war," General Matthew Ridgway, commander of UN forces in Korea, complained bitterly, "this was it."

After the start of 1951, the war largely stalemated, with neither side able to make major gains on the frontline. Compared to the dramatic movements that occurred in the first six months of the war, the rest of the conflict was not relatively as exciting. A poll from January 1951 found that 66% of Americans were in favor of withdrawal and 49% believed the intervention should never have happened in the first place. This was not a popular war that would prompt patriotic chest-thumping and cultural media. And while the US did not have the same levels of failure as what would be seen in Vietnam or in Afghanistan, it was a far cry from the triumphalist narrative that so many Americans were primed for from WWII.

The stakes for why the US was in Korea also was iffy. Sure, the US was beating back the big, bad communists. But Syngman Rhee was far from an ideal local hero to expend blood and treasure for. Furthermore, few Americans knew that the US had been involved in Korea since the end of WWII as an occupational and then support figure for southern Korea. The ongoing social, civil aspects of Northern vs Southern Korean animosity was difficult to pinpoint. Korea was also a particularly brutal war, with atrocities against civilians and unrestricted bombing all too common.

Censorship was also a major factor. Whereas censorship in WWII was coordinated under the largely competent Office of Censorship via Executive Order, which issued very clear guidelines for what reporters could and could not report on, Gen. MacArthur first demanded that news agencies self censor after US intervention in Korea, without official guidelines. Then he loosened censorship rules during the US/UN advance to the Yalu River. When fortunes in the war reversed, he tightened rules again and eventually issued very tight restrictions which stayed in place even after MacArthur was sacked. War information for Americans, as a result, was sporadic, often confusing, and inconsistent because of the scattershot yet repressive censorship rules.

If Korea was not a popular war, why did it not spark the kind of antiwar cultural impact that Vietnam did? While there was absolutely explicit opposition to the Korean War in the states, the Korean War coincided with (and contributed to) a period of broader social, cultural conformity and repression in the United States. Conservatives could employ the specter of Communism and American decline to push back against many forms of "deviance" and counterculture. There were huge costs for persons of influence to speak out against an imagined consensus of hawkish anti-communist foreign policy and the promulgation of one form of an American way of life. Vietnam occurred amid a broader cultural explosion of counterculture, anti-conformity, and broadening of political thought in the US, enabling more of a cultural impact.

For a United States that was transitioning to a state of semi-permanent war mobilization in a global posture against the Soviet Union, the confusing and difficult to swallow results of the Korean War faded from the prominence of American memory.

There are many "forgotten wars" in American historical memory.

The Spanish-American-Philippine War, interventions in Latin America in the early 20th century, American "miniature" wars in the Caribbean and Latin America in the late 20th century, the US troops spread around the globe in the present day, even WWI to an extent...

The Korean War was far from the only "forgotten conflict" that had limited cultural impact in the United States. This is not to say that there's something in American DNA that makes it impossible for Americans to understand anything beyond flashy, sexy conflicts that end in victory. More of to say that there is an ongoing project among historians of American wars and American foreign policy to understand the domestic reasons why some wars seem to spark cultural transformations or have a long-term grip on American imagination, and others seem to fade away. My hypothesis is that it depends on how contemporary US politics references previous wars.

Vietnam has come back to the fore as a relevant cultural lodestone in which to understand the United States because of the experience of counterinsurgency and nation-building in Iraq and Afghanistan. But in the late 1970s and 1980s, administration officials often talked of the need to kick the "Vietnam Syndrome" -- the prevailing sense of defeatism or pessimism about the efficacy of American arms -- through showing success in limited, short, sharp conflicts. The Powell Doctrine which set the framework for the Persian Gulf War was designed largely in reference to the lessons learned from Vietnam. But crucially, the idea was to try to get America to get over Vietnam, rather than evaluate fundamentally the problems of interventionism on a broader scale. Amid the sense of renewed American greatness that, very very shallowly speaking, persisted in the late 1980s, 1990s, and even in the early 2000s, the Vietnam Syndrome did seem to have faded. Still, the impact Vietnam had during its time was tremendous. There are still plenty of veterans around and politicians who made their name speaking out about Vietnam or serving in Vietnam. It remains as a relevant war in which to understand our present.

With Korea, the three years in which most of the fighting happened never had as much of an immediate cultural impact. The US was simply not in a state that was ready to receive, amplify, and digest a fundamentally anti-interventionist message.

For further reading, I HIGHLY recommend the edited volume "Making the Forever War" by Bradley and Dudziak, which compiles together essays that the esteemed late historian Marilyn Young wrote about forever wars. She has quite a few that focus on culture and war, and why some wars seem to stick, and others seem to fade. You'd likely be interested by what she has to say on the topic too.