I just want to say--you can be fully natural and let all your weeds stay and feed the soil, just as nature intended. You won't get much of any food out of that garden. Let's face it--gardening is inherently disturbing, co-opting nature, to redirect its energies into our desired crops. I do lots of weeding, and usually put the weeds in my compost pile so they can work together to decompose, and then be put into my garden where I think the nutrients are needed.
But this doesn't mean I try to remove all weeds. I leave a few butterflyweed, mullein, yarrow and Flower-of-an-hour because they're pretty and feed beneficials. and there are weeds that take over my beds in winter: purple dead nettle reliably in fall and again in March, when it is reliably joined by chickweed and bitter cress, These are weeds I like because they hold the soil and are easy to pull when I'm ready to plant (their flowers also delight the bees and are pretty--the dead-nettle anyway). I would like information about their utility as a cover crop but all I can find is reports from the all-weeds-are-edible-and -medicinal ideologues. The weeds I'd put in the rogues gallery are not the ones named by others, but mostly what comes in on manure (horse nettle) and mulch hay (clover and various grasses). Yeah, clover fixes nitrogen (and I'm fine with it having taken over most of the lawn) but it takes over--I've had to rip out strawberry beds twice after they got infested with clover. The grasses are hard to remove from my clay-based soil without losing lots of soil, and are especially obnoxious when coming up under (both sides of) a fence. I have one patch of flat ground for corn and sorghum, and a rotation of tomatoes. That ground gets tilled, usually once a year, then I put cardboard down between the (relatively wide) rows, with just a little hay on top. Some years it works to pull up that cardboard in September and plant winter rye and hairy vetch, scritch it in with a rake, toss compost atop any patches too hard to scritch in, and get a good winter cover going. In my 30 raised beds (slightly raised, no sides to these beds) I use just vetch or winter peas, as the rye is too hard to remove in spring--but I do plant a couple beds every year where I plan a late planting--the rye dies back if you cut it when it's shedding pollen around the end of May here...and by then it's done lots of good rootmass growth for organic matter. Often I let a bed go to maturity too, in my quest to to figure out a protocol that works for flour production. I also sometimes grow sunn hemp, if I can get seed and get it in early enough--but it dies when it frosts. And I often add daikons if I can get them in early enough, for subsoil work. Trouble with those is, instead of that 12 to 18" root diving deep in my soil, half of it extends above the soil--where it freezes and rots.
A lot of this comes back to that adage--what works for one garden may not fit with another, depending on exposure, climate, soil type, slope and your intentions.