I the idea of “the big ball of goo”. The idea is to launch a fairly large satellite, that when in orbit extends long, hollow, telescoping rods, attached to the ends of which are a very large, spherical tent. The tent is then filled with high expansion, polymer foam.
Gas is then exhausted through the rods to help the ball navigate into the path of the smaller space junk. And even at their high average speed of 15,500mph, hopefully they would be caught by the polymer instead of punching through it. Then when enough was captured, the whole ball would reenter the atmosphere, burning up.
We use aerogel to collect small particles from comet tails so your idea isn’t far off. I would also get Tethers Unlimited on the job.
Using carbon fiber tethers for everything from propellant less maneuvering in space to landing or anchoring to asteroids was at least in part the brain child of physicist and Sci Fi writer Robert Forward. He was one of the founders of Tethers Unlimited and gave very detailed accounts of how the tethers could be used in his book Saturn Rukh. (Who doesn’t love a sci fi story that begins with the sentence, “I’ve got a job for you and it pays a billion.”