Used upper stages seem to be a big contributor to space debris. Wouldn't it be fairly easy and cheap to instead have an upper stage somehow physically lock onto a big piece of space debris and push it down into the atmosphere, or to the graveyard orbit outside of GEO? Sooner or later, the big pieces of junk in orbit will collide and fragment into thousands of dangerous pieces, so removing a small number of the most massive ones should be very helpful in the long run.
Launchers can obviously not always carry a payload which makes use of all their capacity. So there's often place for extra fuel for a small secondary mission after the payload has been inserted in its orbit or trajectory. That's why there's a market for secondary payloads. Even Curiosity's sky crane on Mars had 140 kg of unused fuel after landing. And Chang'e 5T is touring cis-lunar space after having delivered its return capsule to Earth. Upper stages, of course, often end their primary missions in the most commonly used orbits, where it is most important to remove debris. Large maneuvers would generally not be necessary. This concept could be applied opportunistically for launches with spare capacity and in orbits similar to a specific big piece of debris.
What are the problems with this? Is the cost of filling up a rocket to the top prohibitive? Is it dangerous to approach and grab a dead spinning debris object, maybe it actually risks causing a collision and a new field of debris? Or is it an economic problem of the tragedy of the commons which prevents even small investments in removing space debris?