This idea isn't original to me, but I've adopted it as part of my thinking, and combined it with an idea of my own to make things even better.
Point one: to fight an insurgency, one needs boatloads of volunteer infantry, preferably with a variety of language skills.
Point two: America is exceedingly bad at producing boatloads of infantry (particularly volunteers during an insurgency) and is even worse at producing langauge skills.
Point three: America, as the world's foremost military power, will be drawn into insurgencies.
Therefore, what the United States needs to do is take a page from Imperial Britain's playbook. (And Imperial France, but the less we use Imperial France as a role model, the better off we'll be.) The British empire was held together by very few British troops. Outside of Britain's navy, the forces that held the empire together were almost entirely locally recruited. The officers, and essential specialists, were British, but the great majority of the troops were colonials.
Since the United States does not have colonies, we can't recruit directly from them, but we can recruit internationally. Create a military force on the model of the British colonial forces, but with recruits drawn from all over the world, with particular attention to recruiting indigenous peoples and others who speak less common languages. These peacekeeping troops would be trained specifically for peacekeeping/counter-insurgency. They would be armed, equipped, and trained by the United States, and receive all of the benefits that U.S. military personnel receive. On being honorably discharged, they would receive citizenship of the United States and the right to bring their families with them. One could even give them a furlough at the end of their tour, so they would have a month or two, with pay, to get themselves established in the U.S.
This idea would be further improved by adding the baseless deployment I wrote about here
. This arrangement is idea for a peacekeeping force, as they need to be able to move wherever the crisis strikes. (Odds are that we won't have a military base established right where we need it.) The floating warehouses could, furthermore, be stocked with relief supplies, such as MRE's, tents, water purifiers, etc., with which we could also deal with the refugee crises that plague such circumstances.
So what we need is an American Foreign Legion with mobile bases stocked for both counter-insurgency and disaster relief.