Operating under local law makes sense, assuming that the government behind local law has some degree of legitimacy. Local law is to some extent linked to the conflict environment and will to some extent be understood by the people in that environment. A force operating under an utterly remote rule set will certainly be handicapped.
Law, though, is just the beginning of the problem. My own observation of US interventions, particularly those in places not considered strategically critical, is that they are usually designed not for their impact on the target country, but for their impact on the domestic political audience, a circumstance that is not conducive to success.
The US (again in my observation) typically ignores places that are not immediate concerns: there's little effort to develop serious understanding or expertise on environments that are not on today's problem list. When something does break out they are caught flat-footed and there's a mad rush to find some "expert" that will tell the political powers of the day whatever they have already decided they want to hear. At this point the "intelligence" community is tasked with providing a justification for whatever course of action is deemed most salable to the domestic political audience. At the end of it, whatever poor schmuck ends up out in the field is burdened with unachievable goals, unrealistic expectations, inappropriate strategies, and a whole host of other problems.
Unless that changes, and I don't think it will, I'd rather see the US keep it's collective putz in its pants with the zipper well up, and resist the temptation to meddle in places they don't understand. If you can't play by local rules and you aren't there with a clear and realistic objective, better not to be there at all.
IMO, obviously.
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