I would go further with a USFL. That is, while its officers are US military, it should have civilian leadership there as well, for several important reasons.
1) Democrats would always hate them, and a Democrat POTUS and congress would seek to destroy, disperse or abandon them, like they did to the ARVN.
2) They should perform individual missions on contract, with the right to refuse contracts they feel are grossly unfair or high risk. When they do carry out a contract, the *organization* is paid up front, not direct to individual members. And the organization controls the money it has been paid, not the US Treasury.
3) No citizenship agreement at all. Unless they are Americans, this is strictly contract work. They have to operate under bilateral treaties the US has created with other countries, so they will not be under threat from the International Criminal Court (ICC), that would also hate and want to persecute and destroy them. Oddly enough, *not* having US citizenship would bestow a lot of advantages on them.
4) Their civilian leaders, not the Pentagon, and not the US government, would create their Rules of Engagement, mission parameters, and duration of mission. No bureaucrat would be able to impose rules or restrictions on anything they did.
In discussing the proposal of a Foreign Legion for the USA:
“No citizenship agreement at all. Unless they are Americans, this is strictly contract work.”
Gaining French Citizenship for service in the legion has been one of its big draws - as it was at one time for the Roman Legions. Today, gaining US citizenship is relatively cheap, so there is little need to risk one’s life for it - as it became later in the history of Rome, one of the factor’s cited in its decline. Even though citizenship has been a big draw, and a factor in the loyalty/quality of the French force; you can get guys to join for adventure, good training (watch out for blowback on that), to escape their lives (watch out for blowback on that), or just for a paycheck.
But the legal issues are huge and complicated.
Combatants not protected by US citizenship would be subject to the International Criminal Court. US leaders who commanded such units into action, or were simply in office when they committed some atrocity on their own (if the leader were Republican), would be hounded for life by leftist “lawfare” groups, as were Pinochet, Franco and Sharon.
If it is straight contract work, they are purely mercenaries. Rates are much higher for good quality, so they are mostly used for a layer of deniability/covertness, or to get around constraints on the use of regular US military.
Our approach to long term foreign internal defense has been to try to stand up local military and police, like the Iraqi Army and ANA. Local politics and leadership are the long poles for effectiveness, so a different kind of force, under direct US command, with American officers would be new. But it would be vigorously attacked internationally as an illegal colonialist/imperialist violation of sovereignty if stationed anywhere overseas to actually operate. Yet this is exactly what Iran does - covertly and over a long time.
A Foreign Legion would also have to be in compliance with the Geneva Conventions, so legally they would probably (and rightfully) be considered US military. For legal liability (and political liability purposes), it has been seen as easier to work out a deal with another country to use their military, than to try to stand up your own in another country. From that perspective, we have been like a Foreign Legion for many other countries in the world, with our defense umbrella.
Perhaps a Foreign Legion might be a way to more rapidly organize a large force for a surge, so forming a few units to work out the admin issues and to serve as a cadre for expansion in a large contingency, could be a useful addition to our capabilities. Maybe there is a way to finesse it into niches, perhaps as part of UN operations, or even as a UN alternative at some point.
Perhaps the main value of a Foreign Legion, as the poster who proposed it noted, is to outsource the casualties. This is huge from a political perspective (especially over long conflicts), but is also a very significant economy of force measure.