Cover crops need timed out so they don't go to seed. On my garden, which I have been tilling, I mow my buckwheat shortly after it flowers (so my bees get a chance to hit it first), but before it goes to seed. It's a tender annual...so it's done doing it's thing after that and frost will finish it off pretty quickly. Radishes are a very good one too, Ducks4you.
Deep rooted annual grasses you'd probably only want to use when first establishing a no-till plot, because I'd hate to have to pull them up otherwise and they are better tilled under (mow, then till.) It's recommended to double-dig the bed first, but on a big garden, I'd probably get someone to disk it for me. There's some prep work to do to get the soil up to par before you stop tilling it, if you are going to get good results. Then after that you work on continually adding layers of mulch and let it break down slowly.
At planting time, you scrape back enough to get a row of seeds in or place seedlings. Planting diverse types of veggies and rotating them is part of heading off disease problems. Of course you would remove diseased plant material too. A lot of the problems you point out are only problems depending on how you handle them, Seedcorn. You'd want to remove and then compost your healthy plant material and reapply it later as a mulch. When I cover crop my garden now, I till it under and let it breakdown. But since you're not going to be tilling, you're going to have to work with it from a different approach.
The basic idea though, is to disturb the soil as little as possible and not do anything to compact it and keep it covered. I wouldn't have the manpower myself to maintain this on a huge scale. This would be good for a home garden though. But yeah, all literal terminology aside, particularly with your root crops, you are going to be moving that soil some.