Because of economics and lack of political will for conventional armies
First, fun fact: for much of human history, armies were private. So this notion of private mercenaries seems less charged if we remember that this is the status quo ante.
As fatigue of lengthy wars such as the ones you alluded to take their toll on the voters, the relationship between the state and its recruitment base can be strained. Should the US institute a draft, that could potentially result in severe backlash. Contracting private soldiers would bypass this dangerous political juncture altogether.
Normally economic framework can help find the right balance between public/private provision by considering wages, material costs, administrative costs, diversity of tastes and distributional issues. Obviously, things are a little more complicated when we're talking about mercenaries, but this framework is still key.
Essentially, there are benefits from engaging the private sector. The state has a monopoly on force, as it probably should. However, this creates problems. In the absence of competition, Big Green isn't inclined to save costs as it would be in a competitive market. While the mercenary market is not the most saturated market in the world, even a small number PMCs can deliver more on the dollar than state alternatives -- take Executive Outcomes, Blackwater and Wagner Group. Executive Outcomes did not service superpowers like the US in your question, but the point stands that Blackwater did. And it did for a lean green reason.
From the mechanistic point of view, this outcome is natural; soldiers like any other rational market actor, will try to get the most for their productivity. Oftentimes, PMCs pay better. The inconvenient truth is that occasionally all that tax-payer money that went to training your special forces operative is now a free agent. Eventually states must accept the fact that PMCs have some of the best trained units per dollar.