I don't think there is any serious problem with aid supplies in the US; there might be in small localized areas where a single group holds sway but they'd be very much the exception. However, there's a lot of places in the world where the local authorities seize aid supplies and use them for their political benefit. Typically giving priority to areas that support them and withholding aid from areas that don't, in both cases regardless of need. In Indonesia, after the quake/tsunami, there were efforts by local authorities to seize supplies intended for minorities. The Marines and the national government put a stop to that right quick. In Myanmar after the Cyclone, aid supplies simply weren't distributed; three years later, aid delivery material was still stacked alongside the runway, rotting in the sun.
Oh, there are ample problems with aid supplies in the US. Disaster logistics is a herculean effort, and it's very much feast or famine, no middle. In many ways, the biggest problem is when the pipeline really starts flowing, and disgorging stuff at an incredible rate. Donations are a classic example - people give you everything, and I do mean
everything. Theft of aid supplies is often an issue, it's just not the government doing it. But advice on stuff, especially donated stuff, is rapidly becoming "take as much as you want!" because it's less we have to track, sort, store, care for, police and all too often throw away as useless.
The marines very well may be preferable to FEMA. FEMA gives you stuff, and money, up front, and then audits you later, wanting an exact accounting of what you did with it. And then takes back money you can't justify. If your documentation game is not on point, you're in for a world of hurt.
I wouldn't be surprised to see political shenanigans - or perception of same - play out in places like Chicago or New Orleans. And I completely agree on the developing and undeveloped world. International disaster response is a completely different animal.
The only problems with using the big-deck amphibs for aid is that lack the ability to sustain a presence. They don't have the supplies of fuel for their aircraft and LCACs for sustained operations (they're optimized for one or two big lifts and then out.) Likewise, they don't have the stores capacity to keep food and water flowing. Finally, they're slow (20 knots average max) and take time to get to the scene. The carriers can run at 30+ knot and that means they stand a better chance of getting their in the golden hours after the disaster strikes. Also, the CVNs have a lot more in the way of fuel and storage spaces.
I hear that, though my logs hat says lack of ability to sustain presence is solvable; it's a matter of putting the right support in place to keep them operating. And, in many cases, that short supply time may be long enough for critical operations until more permanent structures and logs chains are in place. It's also a lot easier to do when nobody is shooting at you.
I agree on the CVN, though they don't have the vehicle, helicopter transport, and Marines. SO I see them as complimentary assets, not substitutes.