I’m quite aware of the other countries which contributed forces to these wars. What you are missing is the question who shouldered the burden and is still shouldering the burden?
Sixteen UN nations supplied fighting units and five sent military hospitals and field ambulances in Koreaa. Australia was one of the very first to contribute military personnel from all three services. The single largest UN contributor was the United States of America (USA) which at one stage had 140,000 personnel deployed in direct combat roles in Korea. Great Britain, Canada, France, Belgium, the Netherlands, Colombia, Ethiopia, South Africa, New Zealand, Turkey, Greece, Thailand, Philippines and Luxembourg sent fighting units. Norway, Sweden, Denmark, India, Italy contributed military hospitals and field ambulances to the cause.
The United Nations Command (UNC) provided the core military and strategic direction for the anti-communist war effort in Korea. The USA provided the high command for the UNC as well as the vast majority of the logistics, air and naval power, artillery and military infrastructure which made possible the continuation of the war despite repeated reverses suffered due to the actions of an aggressive, determined and proficient foe. The British Commonwealth sent the next most numerous of the UN contingents. As well as Commonwealth air and naval units, the 1st British Commonwealth Division, in which Australian infantry battalions served, took part in some of the hardest-fought battles of the war.
I think you will note it was the US that contributed the vast majority of logistics, etc.
Vietnam, unlike Korea, was not fought under U.N. mandate. It was a war of entirely different characteristics from any other war we had ever fought. Our Military won the battles, but the Politicians gave away the victory. I will despise our so called leadership during this war forever and that includes that sniveling idiot John Kerry who should have been tried for TREASON along with Jane Fonda!
I think it is an important point that a UN mandate existed in Korea but not Vietnam; in the latter we were winning battles for a country that never allowed elections (because “our side” would have lost them).