Unless you're saying that the aid being there in the first place was actively discouraging any self-sufficiency by the sovereign governments who are actually responsible for those areas... in which case I can see why some people (both in the West and the developing world) oppose foreign aid in the first place. Even if we passed the baton around to other countries, the band-aid will still have to be ripped off at some point.
I'm actually not anti-foreign-aid entirely, I think in addition to the humanitarian benefits which I do value, when done well it helps project soft power, giving normal people a concrete counterpoint when extremists like Islamists try to claim "America is an evil empire that hates you and wants you to die." So in that way I don't endorse eliminating USAID... I admit that mainly it was a circus trick to try to hand supporters a concrete W for the vibe of "America First" and/or "Saving Money"