>> literally dig out the whole bed and sift the soil to remove all the roots!
Yeah, fun fun fun. Then replace the soil with something very light and loose, so the next few years of pulling weeds will take as many roots as possible with each pull.
I have a spike bayonet that sometimes lets me pull out a conical core of roots and clay. It will go pretty deep even in heavy clay with pebbles. Then I can "swizzle" it in a circle around the root, like coring an apple. In decent soil, I can lift out a 6-8" long narrow divot. In clay ... it goes deeper than a trowel.
Often I just scrape the surface with a scuffle hoe, over and over, hoping that the weed is less stubborn than I am and gets exhausted sooner (hah!) More likely, something I plant will compete successfully, if conditions are good for "my" plant, and I chop the weeds often enough, for enough years.
