It's really not.
Here's the list of Flying TMs in GSC:
These are
all the Flying moves in GSC: Mirror Move, Peck, Gust, Wing Attack, Drill Peck, Fly, and Sky Attack. Gen 2 added Aeroblast. That's it.
Now, which one of these would you give to a Lv. 9 leader and make it a TM?
Mirror Move is the only reasonable option, but it's incredibly underwhelming, and it's not like "Parrot Mimicry" would fit a lot of mons either way. Mimic was already a pretty mid option in Gen 1, and that was the best version of it by a mile. This would be an even worse copy of that.
To be fair, Mud-Slap is an awful move, but it does work to give birds a fighting chance against all the Rock-types in Union Cave.
If we're being honest, Falkner in general was a bad decision. Nothing about his gym works.