Good! Although incorrect. The Earth has a rotational period of about 24 hours and an orbital period of about 365 1/4 days. Correct would be "The Moon revolves around the Earth."
For physical motions, "rotate" is "to turn around on or as if on an axis" and "revolve" is "to move in a circular or curving course or orbit (the Earth revolves around the Sun)."
That time would be before Theia collided with the Earth: http://www.space.com/25332-moon-formation-history-lunar-evolution-infographic.html
"Вертится" does mean "turns". It's just not the best choice for the more formal language. The singer was going for a more colloquial style. In that case it's not about being childish, but rather "honest", "down to earth", maybe a bit "unsophisticated".
I never said it's a wrong word to use. We use it a lot. It's just not better than "вращается" when we talk about celestial objects. It's just different and fits different style. You wouldn't use it in a scientific report, that's it.