This is a good video on a topic that I've long been interested in: people often take the wrong lesson from the Copernican revolution.
We now know that from a math perspective, any point can be used as the center of the solar system for visualization purposes. The key idea is that placing the sun at the center simplifies some things and facilitates better physical understanding. And in any case, the sun moves too.
re: "what new ideas are facilitated by viewing things in this way?"
I find that this line of thinking helps a lot, at least personally, with being mindful about confusing the map for the territory, i.e. does this level of abstraction help me understand this particular function better vs. this other level of abstraction that might be more accurate but is encumbersome to facilitate insight.