> If you want to rebuild your soil, let the weeds grow and then either use them for compost which you'll later top-dress your soil with, or just chop them and leave them be for the cover crop effect, or work them in slightly, up to you.

This has been my primary approach. Letting the grass and "weeds" grow relatively tall then trimming them down every month or so, letting the clippings rest where they are more or less.