Hawks are protected under Migratory Bird Act - and there are very specific exceptions for keeping raptors for sporting use. Prior to the regulations you could just go out and pluck one from a nest and have at it. Many birds these days are captive bred - especially the rarer ones, such as falcons.
Also, to ensure you learn the craft and caring of birds, there is a required mentorship program where you need to serve under a registered falconer. Makes sense.
No one ever said regulations of sporting birds vs recreational weapons makes sense.
Yes, and while some species are doing well, population-wise, others are rare (usually due to habitat loss or restricted range) and the 'take' of wild birds is heavily restricted. There are federal restrictions on some species that override state preferences. So, for example, there is no allowable take of the highly desirable aplomado falcon (too endangered), which must be purchased from captive breeders. In regard to peregrine falcons, the feds set overall take numbers, and then allocate a specific number of permits for each state within the falcons' range, depending on the state status of the species. Still other species are not restricted federally, but states restrict take depending on the health of the state's population. So white-tailed hawks, black hawks, gray hawks, etc., cannot be taken for falconry in TX, but lots of other species can be as long as you have the proper permits, ranging from very common species (e.g. red tailed hawks, Cooper's hawks), to slightly less common (e.g Harris' hawks, ferruginous hawks), etc.
But the short answer as to why it is so heavily regulated is the historical mass persecution of raptors (which is still going on illegally, though nowhere near at the historical level) and also I suspect it has to do with the fact that it is much harder to keep and train and successfully hunt with an heretofore wild animal than just picking up a gun, aiming, and pulling the trigger.
ETA: not that I understand why anyone would bother flying black hawks or gray hawks or broad winged or red shouldered any of the other fish/frog/lizard/small mammal specialists, but inevitably some crazy falconer will try it just for the novelty LOL.