The United States relied heavily on proxy forces during the Cold War. Millions of Koreans and Vietnamese fought alongside U.S. troops. And Soviet personnel were involved (secretly) in direct combat against the USAF in both Korea and Vietnam.

In Latin America, Africa, Europe, and Asia, the U.S. government trained and funded armies and militias that fought on its behalf. Early U.S. intervention in Vietnam followed a script rehearsed several times in Latin America; provide cash to a friendly elite, train intelligence units on the dirty work of suppressing dissent, furnish quiet military assistance (signals intelligence, logistics, occasional Special Forces raids or deniable air support).

I think the key difference for those two conflicts vs. other cold war proxy conflicts is China. Both of those conflicts bordered China directly enabling China to use its primary military weapon, overwhelming manpower.

China and the Soviet Union armed North Korea for a quick campaign to seize the Korean Peninsula. They were well equipped had the element of surprise and created such chaos in South Korea that their campaign would have succeeded without US intervention. The US counterstroke was decisive and threw the North Koreans into full retreat. It primarily used South Korean Troops backed up by US troops. In November of 1950 it looked like the US was going to unite Korea with relatively little loss of American lives. Then China entered the war and changed it from a large military operation to a full scale war.

In Vietnam the US couldn't put a decisive end to the conflict because they couldn't attack Hanoi with ground forces without risking China entering the war directly on the side of North Vietnam. The US inflicted nearly a million casualties on the NVA and Viet Cong with only 55,000 US casualties. Most of the casualties we inflicted didn't come from rifles they were done with bombs rockets and artillery. It is possible that if there had been a way for South Vietnamese troops to go on the offensive supported by US equipment and a few US troops they might have been able to take the north and end the conflict. But With China in the way a loss for the US and South Vietnamese was inevitable. It was a war that would only end if one side achieved victory and the US had no way to achieve victory.