Why do the planets revolve around the sun on a plateau and not as electrons do?

Answer

The planets all move in roughly the same plane around the sun, because they were born together in a disk of dust and gas, which is a byproduct of the sun’s formation, and which revolved in the sun’s equatorial plane. I have already described that creation scenario on this website, see e.g

www.ikhebeenquest.be/academic/question3.jsp?id=29338

But the picture you show of those electron orbits is less evident. It’s not a picture, by the way, but a performance. The analogy between electron ‘orbits’ and planetary orbits is somewhat misleading. We do know that different electrons in atoms occupy different levels of energy, but pinning them to a well-defined orbit is wrong. But there is indeed no reason to believe that electrons should be located (to the extent that an exact location makes sense for them) in some equatorial plane of the nucleus. In this sense, there is no preferred direction in an atom.