> This doesn't actually work. If you push something down in orbit, you just make its orbit elliptical.

Came here to say this as well! The only caveat to that I can think of is if you're already low enough in orbit, the push making the satellite's orbit elliptical could put it in a drag zone, and then air would do the slowing down for you.

Also, thank you Neal Stephenson for introducing me to the wonderful would of orbital mechanics via Seveneves!