> What particular convenience does it provide for the US government to do it there vs on the US side?

It reduces legal accountability (I know the US courts have generally exempted border operations from the constitution anyway, but that interpretation could change in the future) and makes it easier to prevent people from e.g. landing and claiming asylum (yes there are measures to penalise airlines and oblige them to return passengers, but they're not always fully effective). More subtly it means there's less pressure to have reasonable border rules, since turning someone away before they board is lower-stakes. And having an official, pseudo-law-enforcement presence in a country is valuable almost in itself.