Oh, you won't ever get it all, the idea is to clear a safe lane. Aerogel casing, ...lower mass, less energy to change orbit, and will stop a speeding bullet. The idea would not be so much to get rid of it all, but even as something which could shadow and protect critical craft it might be good.
No such thing as a "safe lane," either. At best you could decrease the density of the debris some. The thing is, the logistics of an object such as you describe would be daunting. To have any reasonable (and still tiny) chance of intercepting debris it would need to be huge, and it would have to be reasonably dense and therefore heavy. Good from the standpoint of staying in orbit, and absorbing impacts, but not so good from the standpoint of trying to launch/assemble something big enough to be useful.
Supposing it's up there: there's no point in trying to capture anything. Small stuff might well vaporize on contact. For bigger pieces, if something blew through the aerogel it might slow the debris down enough to lower the perigee of the debris into deeper atmosphere, and that would cause the debris to burn in sooner.