This is based on very limited real world experience, so take it FWIW -

-There seems to be little debate or disagreement about the need to address death\suffering by any and all means. That may mean very basic stuff, but, particularly in the context of an urban insurgency, may be much more complicated.

-For a while there, while the Army was trying to (re)learn COIN, it seemed like the Army was thinking of "hearts and minds" and "warm and fuzzy" as interchangable concepts. I don't think that's appropriate, and it seems like the Army has, in large part, moved past it.

-At a low level - Being friendly to the locals, respecting customs, etc is appropriate from a standpoint of basic respect, human decency, and promoting interaction. More elaborate stuff should probably be weighed carefully against the benefits. Dropping a couple of million on soccer balls to hand out to the kids (or whatever) should probably be approached carefully. If it facilitates interaction, HUMINT, whatever - go for it. If it's just an effort to look like nice guys and gals, a little skepticism by the commander is in order.

-Massive infrastructure projects are, overall, best handled by non-military agencies. But as wm and others point out, the US's agencies for doing that have eroded. Perhaps not in size, but the type of stuff that USAID could do (directly) in Vietnam appear to be much more limited today. This appears to be turning around, but it is, at best, a work in progress. My understanding is that the UK has some of the same issues. Accordingly, as many have pointed out, the military jumps in. The unity of command provided by that approach is also an issue. Also, while US military units never lost freedom of manuever in Iraq, I think you can make a case that civilian agencies did, making a whole of government approach difficult. Probably not the case today, but impressions may linger. Lastly, the US, in particular, has substantial funds available to military commanders, with (relatively) few bureaucratic strings\nightmares attached. Those funds have, at times, been used for massive projects. That might or might not be wise (see below), but part of giving a commander more resources and lattitude is increased liklihood of mistakes. We have wisely resisted applying a lot (in relative terms) of oversight to CERP.

-Large scale infrastructure projects may be a subtle, low-coercion form of population control. It's sufficiently different from what has traditionally been called "population control" (as Ken and Wilf have pointed out elsewhere) that perhaps that not the best term. But a school that redirects children from Tribe A through a path through a dangerous area occupied by Tribe B (less violence, fewer revenge killings), extends the reach of the local government (gotta pay the teachers, gotta patrol the area), employs a few locals and counteracts an insurgent narrative might be a great idea. Our ability to predict 2nd and 3rd order impacts is limited enough that I'm skeptical of leading with massive projects that aren't handed off to USAID or a similar agency, and "schools" in particular would give me pause - but we shouldn't get wrapped around the axle about this or that TTP. If a whole of government approach isn't working in a given security environment, then the military has to step up until the other agencies get ready. Others have talked about the military's efforts giving license to those agencies to sit on their hands, but that's another discussion.

-Large scale infrastructure projects may also be a form of large scale national level diplomacy. They may also something similar to bribery - nothing wrong with that, but apply carefully and evaluate for sustainability. They may also be a counteraction to an insurgent narrative. Or they may just be a bad idea. In short - I'm skeptical of one size fits all answers.