If I said I can fly by flapping my arms quickly I would assume the burden of proof would be on me to prove it as opposed to others proving that it's impossible
The argument here is more like: people can’t fly by flapping their arms, therefore human flight is impossible.
The more analogous claim would be "it's impossible to flap two arm-like appendages quickly and achieve flight," and the burden of proof would indeed be on you for making that claim. (Of course, in that case it would be easy to disprove by pointing to birds or even with computer simulations and toys if birds did not exist).