Why not mount some sort of thrust engines on the asteroid to allow control and adjustment the orbit?
Well, because this isn’t like a tiny low mass satellite that only requires occasional adjustments to keep its orbit from decaying. This is a high mass satellite with an atmospheric anchor continually dragging it down that would require massive thrusters constantly burning lots of fuel or it would decay past the point of no return very quickly. Where is the fuel going to come from to do that? Just constantly launching rockets from earth to ferry a little bit of fuel per launch up to the asteroid at billions and billions of dollars per launch?
Who will want to pay for that? Who will want to take the risk that if a few launches go wrong the thrusters run out of fuel and life on planet earth possibly goes extinct?