Apparently the conquistadors noted the Aztecs using trained hawks, so some parallel development seems plausible.
We did get rid of certain items deemed hazardous: the screen perch, which really requires constant supervision because raptors can hang from it and not be able to get back up, and single-piece jesses were replaced with a 2-part system where the jess part is removable. Should the bird get lost it can take the jesses out and not be hampered by them.
The main innovation is radio telemetry. Bells are still handy for immediate feedback of your bird's location, but telemetry is fantastic for falcons which can fly 20 miles away in a half hour if they feel like it, or hawks that will sit tight in a thicket on quarry and be completely invisible.
(And what did you fly? A pleasant surprise to meet another falconer on HN. I flew Harrises and various accipiters. Lost my last bird in January to a wild peregrine and am considering a female sharpy for my next bird.)
Also, I like the bear suit.