it just goes against everything I've known about gardening since I was a kid.
Me too - my dad taught me all about spraying the round-up, and the malathion, and the seven dust, and all the good leftovers from the bombs we didn't need between the big war and the little ones that helped keep the fertilizer manufacturers in the bomb business. Or the bomb manufacturers in the fertilizer business - whatever.
I guess the thing to get past is that you don't need nutes therefore there's nothing to compete for. The flowers, radishes, peas and such that you can use for a cover crop can be selected for the bugs they do or don't attract. Peppermint is a great groundcover that runs like strawberries and deters pests. So when I say 'weeds' I mean stuff I am not eating.
The fact that the cover crop is a bio-accumulator they are helping unlock minerals and such that are in the soil, then when mowed they return whatever they used to the soil. The goal is to turn your garden into a big worm bin. You can constantly add mulch, but grass dries up and blows away, or clumps up and molds. If it works perfect, the worms will demolish it in a week or two and you have to add more. The last thing I fed them was ground up malted barley just dry topdressed. I also sprout barley (or corn, or alfalfa or whatever) then put it in a blender and toss it in a 5g of water. That will feed my 10 x 20 garden with a lot of enzymes and stuff that get the soil working and the plants dig it. Compost is good too.
I am learning about this as I go so I'll post the bad stuff too. I planted too early and lost a lot to a late frost. But using my weed-learnin' I let that shit re-veg and it's all putting out good new growth now. I think if you picked a corner of the garden that doesn't usually do as well (we all have that) and try it with just a selected cover crop you might be surprised. I may have all my stuff get eaten by bugs and get nuthin' who knows LOL