I draw a blank. I learn from Wikipedia that the US had conscription during the Korean War, but it seems to have been uncontroversial, or at least nowhere near as controversial as with Vietnam. The war reached a stalemate before The Wild One and Jack Kerouac's On the Road etc, at a time when the counterculture wasn't yet a mass movement. I have the impression that the media in both the UK and US was still willing to toe the government line in the immediate post-WW2 years. Furthermore Korea was generally perceived as a "good war" along the lines of the Falklands conflict or the Finnish Winter War - the politics were easy to grasp, both sides wore uniforms and in theory fought under the rules of war. There were "goodies" and "baddies".
I note that the US armed forces suffered almost 34,000 deaths in the three years of the Korean war, versus less than 60,000 for almost a decade in Vietnam, but again I have the impression that people in the early 1950s were less fazed by high casualty figures after a massive war that had killed hundreds of millions of people.
And of course it was at the time overshadowed by the Second World War. They were still making WW2 films in the early 1950s and for many years afterwards. And Korea was just before the widespread adoption of television, so footage of dead GIs wasn't broadcast into people's living rooms, not that it would have been anyway. And to top it all the generation that fought in Korea didn't tend to talk about their experiences. They were silent. There was some grumbling a few years ago about the US army's treatment of Korean prisoners, but I think the general consensus is that the Koreans were no better.
It has to be said that during the 1980s and most of the 1990s the Second World War was almost a forgotten war as well. After A Bridge Too Far in the late 1970s there was a dearth of WW2 films right up until Saving Private Ryan in 1999. My hunch therefore is that a really good film about Korea might have revived interest in the war. As you point out MASH was huge, but didn't feel Korea-y. Basil Fawlty was a Korean War veteran. That confused me as a kid. In one episode he talks about fighting in the war, but he was obviously too young for WW2. I just wanted to say that.
The paradox is that Steven Spielberg, George Lucas etc were inspired by the WW2 films they saw when they were kids, so in order for one or other of them to make the definitive ground-breaking Korean War film they would have had to grow up with Korean War films, which wasn't going to happen unless Korea was already popular, which it wasn't.