Not all cats have a problem with flaps.
I have two cats they have zero problems with the flaps. They sometimes use the flap as a short cut to sneak up on the other one when playing.
I have those electronic ones you register with their microchip so only they can use the flaps.
But, I still get gifts like mice, skinks, praying mantis, cicadas, leaves, etc. The flaps don't stop gifts.