Obviously, if there was a significant atmosphere the moon would experience atmospheric drag and would fall to earth.

Would it?

The Moon is slowly drifting further away because Earth’s rotation is significantly faster than the lunar orbit. Some of Earth’s rotational energy is bled into the lunar orbit, causing it to increase in height, which in turn lowers its orbital period.

If there was a significant atmosphere for Earth at that height, you’d think it would actually give the Moon a boost