Because they have to eat a lot. That's why they fly from tree to tree. And on the way to the moon there is no place for them to stop to rest, and more importantly to eat.

Some birds eat worms and seeds on the ground and some eat insects and some eat fish. If you want birds to fly to the moon you need insects to fly to the moon so the birds have something to eat, but I don't know if they can eat while flying or have to catch insects when they are asleep. Obviously the ones who catch fish can catch them moving, but water is different than the atmosphere and fish eating birds are different than insect eating birds. Their mouths are for scooping instead of pecking.

If you could design like a cage that the bird could carry with a feeder and a perch, he could fly, then coast and perch and eat, then fly some more and possibly reach the moon, but then he would be trapped in a cage and if there were a latch to let him out he might not be good for the moon's ecosystem.