Indeed, one would expect that (hard) airfields do have some repair facility, but you can't make it a hex feature, not only would it require a lot of work, but there are the carriers too, which are units. One size doesn't fit all.
The easier solution (IMHO!) would be to simply create something similar to the actual system where transports and fortifications have the option to repair ("heal") adjacent land/sea units: Just add a second option (second checkbox) allowing to repair air units, so people can create ground repair units
and air repair units.
That way scenario creators can place a "repair facility" adjacent to any airport which would be supposed to be able to repair planes (be it a building or a mobile unit).
Remains the problem of airplane carriers, which are neither transports nor fortifications? Not really.
Think of it, where would a carrier far, far away from home get that infinite supply of spare parts from? Well, obviously from an optional
supply ship! Which can be a transport... Just place it alongside your carrier, and all planes inside that carrier get repaired.
So, the solution would be to just add a second version/layer of "healing" to transports & fortifications, this time affecting air units.
That would solve all scenario issues (I can think of) like situations precluding repairing (no resources, supply route cut, etc.), and, most of all, it would avoid making damage to planes totally irrelevant (just leave them a turn on any airfield and they're good as new again).
The problem being that, unlike ground repair trucks (I am a big fan of) which are rare, always too far, or have just moved or whatever, planes will
always eventually land on an airport/carrier, which will then
always automatically repair them. We can't have that, because it means that unless a plane unit is totally destroyed, it will be fully operational again in a turn or two, no more fighting with severely weakened units because you've got no prestige to fix them anymore... We need to be able to limit air repairs, to keep the player on his toes.