I’m a student of Dr. Elaine Ingham’s Soil Food Web. Learning what to do with what weeds, and more importantly, why they’re even there in the first place, is enlightening, to say the least. Weeds are part of “plant succession”. Prior to Dr. E’s teaching, I’ve been no-till for ten years. I mulch with leaves and straw (whenever it’s free) and my soil has NO compaction (tested with penetrometer). My 1500 sq ft garden has almost ZERO weeds, and I always attributed it to the mulch and spending five minutes a week on a weed walkabout, removing any that dared poke their head above the soil. Without trying to elaborate on her entire lessons, weeds proliferate in disturbed and fertilized soil because that provides Nitrate... weed food...having proper no-till practices and healthy soil biology provides ammonium... plant food.
In general, she teaches to leave weeds be as living mulch, unless they’re going to outcompete your crop for light. When annual weeds need to be removed, they are always cut, never pulled, leaving the roots... and carbon, in the soil. The one weed I am diligent about keeping at bay is Scotch/Twitch/quack/crab grass... whatever it’s called in your area. It has to be gently teased from the soil or it will regenerate from every broken root. Even noxious perennials can be killed by cutting instead of pulling, but one must be diligent to keep the green chopped off... eventually the root starves to death!