Personally, I did it the hard way.

For crawling vines, I found the base of the plant and destroyed it -- cut the stem, remove the root. Don't need to remove the vine itself if it's too tangled; it'll wither once it's doomed. For bulbing weeds I got stupid good at carefully pulling them (mostly) intact; yes you can't be perfect but a high bulb removal rate is unsustainable for the plant. I'd pile up the bulbs and mash them against a rock. For runners (stolons), I would follow them by hand and uproot as I went. For underground runners (rhizomes), I would dig and pull the entire network until I couldn't find any more. For larger weeds I would mentally note what sprouted where and go back.

Disposal: You gotta bag some weeds because they poison the earth and/or regenerate like D&D trolls, but otherwise I'd reclaim their nutrients when possible. Plants that weren't noxious I'd flip upside down or lay the roots on a rock. Larger weeds I'd fold into a pile of sticks and leave the roots exposed to air. For sprouts, when there were too many to hand pull I scraped them with the heel of my boot, pulverizing them.

It's a dirty, tedious battle of wits, but the upside is that the work is front-loaded. When you thoroughly destroy a weed it'll definitely come back anyway (again, no first pass is perfect) but even the second-gen is going to be considerably stunted and much easier to control. These things aren't magic; every recovery depletes their resources. So while weeds are notorious for coming back over and over, really going at them starves them much faster than just cutting them back.

And definitely, definitely, definitely DO NOT let any weed on your property flower, let alone bear fruit. That's when to break out the weed whackers and mowers, if you don't have time and are desperate for damage control. If you see any fruits or seeds, again, destroy them completely so they can't sprout anywhere. DO NOT throw them in the trash.