Is there any reason to believe that precious metals are found in greater quantity in the asteroid belt? Maybe a bit more accessible?
It is likely that many asteroids contain large amounts of precious and rare metals that would not need to be sifted and sorted from large volumes of worthless material.
As others have pointed out - it is an interesting problem in that recovery of the material would be very expensive given current technology. People will certainly try though and technology will improve so it will be feasible at some point. However, this might present another problem - as a “rare” material becomes more prevalent the value may decrease unless they are materials for which demand increases (like rare earth metals).
A cheaper and emerging industry in the future will be deep-earth mining as large amounts of rare metals are known to exist far below the surface.
It is also likely that some will oppose it on legal or environmental grounds - there may also be a “claims” process on the part of nations and/or corporations. These will present some interesting legal cases!