Astrophysicists say everything is flying away from each other, making everyone's perspective seem to be the center of the universe. They draw dots on a balloon and blow it up to show them becoming more distant.

Baloney!

Everything is shrinking. Molecules are falling into themselves, the spaces between them getting closer as gravity increases in strength relative to electromagnetism. Things look further away over time because they are getting smaller. I'll draw two eyes on an inflated balloon and slowly let the air out. As the eyes get closer together things look further away.