The US interference in Latin America was not something that stemmed from the Cold War. Long before communism was even a political ideology the US established the Monroe doctrine in 1823 that stated Latin America was under the US sphere of influence and that they would not tolerate influence of other powers in the region. What this entailed was that Latin America were in a sense satellite states to the US that served their political interest and the US intervened in their internal affairs however they saw fit. The US has done this multiple times prior to the Cold War such as pushing for Panama to succeed from Columbia so the US could construct their canal there and the Banana Wars where the US essentially invaded Honduras to ensure their economic interest were secure which led to the term Banana Republic. Thus, the US had a vested interest in Latin America long before their ideological conflict with the Soviets even erupted.

When the Cold War did finally kick off, two central characteristics of it was the ideological competition and competition for sphere of influence between the superpowers. What this entailed for Latin America in specific is that the US wanted to maintain its control over the region as the governments there (installed by them or not) were friendly, supported US interest and were economic partners for the US. In the case of Cuba and other nations in proximity to the US, they were strategically important as their proximity was a potential outpost or launching off point for hostile forces against the US. The conversion to communism in the region would compromise strategic and economic concerns as it would create hostile regimes to the US and undermine their prestige in the ideological conflict. At the time it is important to note that the prevailing belief was that communism was a threat to western civilization and their way of life due to its atheistic nature and its for rule of the proletariat. It was not necessarily believed to be inferior because it did not work and at this point no communist state had yet collapsed with one of the world's superpowers being communist thus providing them a level of empirical legitimacy as a political alternative to democratic capitalism. A country becoming communist would not necessarily mean their eventual collapse to the policy makers of the time so to let them turn communist and hope they collapsed later on was a foolish gamble given the strategic implications.

The US at the time also subscribed to the domino theory which in short believed that if one country fell to communism their neighbours would follow suit eventually taking over the whole region. While the validity of this concept is debatable since there is merit for both sides of the arguement, the important thing is that the US believed it to be true. Thus one country in Latin America becoming communist to them had the potential to spread the revolution to the whole region which would turn them against the US. These nations were important economic partners to the US and as stated above were strategically important as well. Thus the whole region falling to communism would be an unacceptable situation and they worked to combat it, enforcing the new Truman Doctrine of containment using the precedence of the Monroe Doctrine. The proxy conflicts themselves effected the prestige of the US which was one of the measures it used to compare itself to the USSR. Their ability to protect their allies or sphere of influence reflected on their standing as a superpower. Thus to let the states adopt communism (which was often through revolution) would be seen as a defeat for the US as their allies would be ousted and a victory for the USSR it implied the superiority of communism over the old regime.

Overall, the while you could make the argument that the eventual failure of communism would show the superiority of the capitalist-democratic system, it simply was not in their interest to do so and went against precedent. Letting countries become Communist, or even anti-US for that matter would have had strategic and economic implications for the US that they would not just accept. At that point communism had not yet proved itself to be an unsustainable economic system so to hope that they collapsed would be wishful thinking and the act of communism becoming dominant in a country replacing the old capitalist system undermined the claim that it was inferior in the short run which in turn hurt the prestige of the US. This argument only make some sense with the benefit of hindsight and doesn't take into account the immediate short run implications letting states turn effectively hostile.

Edits for clarity and better phrasing