To show just how complicated things are, though, the previous government of Afghanistan had negotiated a deal with a UAE-based group to operate the airport and “[t]he Taliban will need to cut a separate deal with them.”
The deal gets Turkish President Erdogan a bargaining chip with the West, and also helps avoid Turkey being the recipient of refugees (as they are from Syria) and gives them control of the drug trade out of Afghanistan. Put another way, Turkish control over the airport probably makes continued expatriation of Afghans possible while also being a bad result in greater Turkish influence in the region as well as inside NATO.