This doesn't actually work. If you push something down in orbit, you just make its orbit elliptical. After half an orbit it will be at the same altitude as you again, and after a full orbit it will hit you from the top. If you want to lower its orbit, you need to push it back.
Orbital mechanics is weird.
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!
The Kerbal never got back to his ship before running out of propellant.
if you want to lower its orbit, you probably need to take a lot of complicated stuff in to account which doesn't fit into a comment. i won't charge the orbital dynamacists implementing my idea more if they replace "down" with "the optimal direction for this target, at the optimal moment".
mass of the target, cross sectional area of the target, any out gassing it might be doing, where we're trying to maneuver to next, etc, etc.
if the optimal direction ends up being exactly "down" or "back", i'll buy you a beer.
You may be to do this with atospheric drag if you add enough ellipticity but they would only be lowering the orbit indirectly.
you know, in the wildly hypothetical world where this was implemented, i don't think the folks cutting the check would mind if you deorbited things directly, or indirectly.
meet up with the debris, reorient yourself and the debris such that the debris is closer to the earth (reaction wheels, magnetic torquer bar and patience is all that's required), and then just mechanically kick it down (store up energy from the sun and then trigger a solenoid). the debris itself becomes your propulsion. you get a tiny boost up, perhaps towards your next target, and it gets a (admittedly!) tiny boost down.
obviously it doesn't instantly dispose of everything, but progressively pushing material further down the gravity well is the next best thing.
not claiming this would be cost effective.