One of the main reasons they aren’t doing it is that the fairing halves are so far out to sea (about 800km down range) that no helicopter could have the range with a fairing dangling down to make it back to land (the Sikorsky Skyhook, for instance, only has a range of 200nmi or so... maybe you could get a specialized helicopter to do it with heroics like drop tanks, but the cost would be much higher), so they’d need to land on a ship after capture anyway. By catching the fairing halves with ships directly, they save the substantial costs of also having helicopters.

Notice that with RocketLab reuse, the helicopter is staged from a ship at sea. However, probably can save more money by recovering a RocketLab Electron first stage than half of a fairing (you’d need one helicopter per fairing halve but only one helicopter for Electron... Electron also has a much smaller and less awkward aerodynamic cross section than a F9 fairing halve). See: https://m.youtube.com/watch?v=enndCzvZpZk