First I've every heard of any such proposal. How exactly would you go about stabilizing every last decay product to the level of background radiation?

Your statement is equivalent to noting that there's nothing in chemistry which prevents us from building coal-fired power plants which capture all CO2 and toxic pollutants (mercury, sulfur, particulates, NOX,, radioactives, etc.) as well. However it's economically and thermodynamically infeasible. Not that emissions haven't been drastically reduced from early designs, but it turns out that that's still not good enough.