I'm not trying to make a point that the air carries objects along with it, even though that's generally true. I'm saying that the air, objects that happen to be in the air, and the ground the air is above are all rotating at effectively the same speed and direction, and gravity keeps everything curving back toward the ground, and so there's no constant energy required to keep from falling off the earth.

The air is rotating with the Earth too, and there's no reason for it not to on average because it's the lowest energy state.

Speed is always relative to another reference frame. If you're riding in an airplane and eating a peanut, you could say the peanut is going 400mph and it would be true relative to the ground, but it's not going 400mph relative to your mouth. Relative to the surrounding air, maybe it's going 380mph if there's a 20mph tailwind. Relative to the sun, the peanut is probably going a bajillion mph. The only energy/power required is for the plane to overcome the aerodynamic drag from moving relative to the air, and the constant 9.8m/s^2 acceleration upward to counteract gravity.

Said another way, there's no energy required to maintain a constant rate of rotation for matter that's held together by natural forces in a closed system, only momentum. The Earth overall is largely a closed system, as it is moving though a vacuum. Over cosmic time scales, though, the Earth does lose momentum because it's not a perfectly closed system. Days are becoming slightly longer from year to year. Some of that is due to tidal forces with the moon, for example.