The moon doesn’t actually orbit earth in thr sense one would think.
Both earth and the moon orbits their mutual barycenter.
This point is – because earth is alot bigger than the moon) inside earth. But not exactly at its center.
For this reason earth’s rotation wobbles a bit.
This is what’s causing the tides.
Its not the moon pulling at our atmosphere.
Here’s an animation that makes it easier to visualize.
https://en.m.wikipedia.org/wiki/Barycenter_(astronomy)#/media/File%3AOrbit3.gif
So the answer to your question is that the moon isn’t pulling away the atmosphere because the atmosphere is much more attracted to earth than to the moon.
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