Well, it's complicated. Simplest explanation: the U.S. preferred to do so through proxies or through military aid. (Quick note: the United States didn't invade Vietnam, at least from a strictly legal perspective; it participated in defending the South Vietnamese government, and while the North Vietnamese saw it as a kind of foreign meddling, U.S. goals were limited to preserving South Vietnam).

Disclaimer: Because I'm not very proficient in Congolese history, it will not be discussed below. However, the Congo Crisis is a significant Cold War episode with U.S. involvement, and in 1964, the U.S. does back mercenary contingents in the country to win support for Mobutu Sese Seko.

First of all, don't overestimate the extent to which the Soviets and Chinese were supporting revolutionary movements. The FLN in Algeria attracted surprisingly little support from the Soviets, for example, because the Soviets were more interested in making inroads with de Gaulle. They also weren't working together: in both Rhodesia and Angola, the Chinese and Soviets backed factions not necessarily inclined to cooperation with one another.

Secondly, don't fall into that trap of thinking of Africa as one great big entity which can be neatly spoken for. Most of West Africa doesn't really participate in the Cold War, and to the extent that it does, it's over economic aid/development (and honestly, the Soviets play a very limited game there). Apart from the Congo, Central Africa isn't affected. Egypt is important in the Cold War, but more as a Middle Eastern power than an African power (Egypt sort of straddles both regions, both literally and figuratively), and Libya and Gaddafi are significant later on, towards the end of the Cold War. That leaves East Africa, where the two most important countries in the Cold War were Ethiopia and Somalia, and Southern Africa: South Africa, Namibia (South West Africa at the time), Zimbabwe (Rhodesia), Mozambique, and Angola. (I know I'm omitting Guinea-Bissau and the Cape Verde islands; for my own ease, I'm lumping them in with Southern Africa).

To your question, western powers needed to be sensitive about the issue of colonialism. Remember that this same period sees Africa and Asia gaining political independence. Now, the United States is more inclined to work with colonial powers for the sake of stability, but that can only go so far politically. Portugal maintained a pretty transparently racist empire that leaves black Africans as second-class citizens, and Rhodesia followed suit. Portugal did receive some aid from the United States (through NATO), but out and out military intervention is simply out. For one, the United States is itself tied down with Vietnam. Yet even if they hadn't been, it would have been a catastrophic propaganda defeat to fight an insurgency on behalf of governments that were white supremacist, and it likely would have invited more Soviet/Chinese aid (not to mention domestic backlash). And in the wake of Vietnam, most Americans didn't want another Vietnam; these were precisely the arguments deployed in 1975 and 1976 to try and prevent any U.S. support of UNITA in Angola or for support for South Africa's incursion into Angola.

For a long time, the U.S. also didn't need to become directly involved. Portugal waged increasingly high-stakes counter-insurgencies from the early 1960s until the Carnation Revolution in 1974 that were aimed at maintaining their control of their empire; the insurgencies they faced were frequently Marxist-led or inspired. South Africa faced international condemnation because of apartheid, but the U.S. still saw the country as a reliably anticommunist ally in the Cold War. South Africa became a regional bulwark that could support the others, which it did, providing police and arms to the Rhodesians. The Nixon Administration's National Security Memorandum #39 held that the white minority governments were there to stay, in one form or another, and so it was better to treat with them.

But then in East Africa, the U.S. supplied a considerable amount of military aid to Haile Selassie's government. Once the revolution removed Selassie from power and placed the Derg in control of the country, U.S. aid petered off once the country drifted towards the Soviet Union. Somalia had been a Soviet client state, but the Soviets were more interested in Ethiopia (bigger country, bigger ally) and swapped to backing them. Somalia (for reasons unrelated to the Cold War) invaded Ethiopia, which then attracted the support of the United States and leading to war between Ethiopia and Somalia. U.S. (and Soviet) military aid was significant here, and the U.S. backed Somalia until the late 1980s, when Siad Barre began losing control of the country and they were no longer a necessary ally.

Once the white supremacist regimes were gone (minus the South Africans), then you start to see closer U.S. involvement in the southern African struggles. It's easier to back UNITA in Angola than the Portuguese because they don't have the taint of white supremacy, and while the Reagan Administration couldn't directly do so until 1985 because of the Clark Amendment, they nevertheless worked to find support for UNITA. It was more difficult to supply aid to South Africa because of the levels of international condemnation and the arms embargoes that were put in place following the Soweto Uprising in 1976, but there are some arguments in the book Apartheid, Guns, and Money that there were back-channel connections with plenty of western powers and arms companies that South Africa could still get things, albeit at greater prices.

Can I piggyback on this question with a related question? How come the United States supported the Khmer Rouge? Pol Pot was a monster, killed a bunch of people and set up something closer to a Communist-style totalitarian regime.

However, even after the Khmer Rouge were ousted from power by the Communist Vietnamese, the US supported the Khmer Rouge as the nominal leaders of that country--despite the fact that they had no real power and were living in exile.