Hey Rocket,
Congrats on the natural progression and hike in quality!
Myself, I dont actually rip them out when no tilling, unless you're after the CBD in the roots.. which is something I just learned about.
But I do chop em, and leave about 4" of stem usually.. then 10 days later or so, after worms and bacteria and fungi do their thing for a bit, among other creatures, the stems you left will rip out with ease, without disturbing the surrounding structure and micro environments. This is much better if you want to establish a no till ..
Cover crops.. how cold do you get? If you can't grow greens there may be a couple other options..
I usually use comfrey, heirloom peas (to fix N), and or borage. They make great mulch at first then melt into the soil, keeping the organic content and nutrients up there. With Comfrey, which can have a 14 ft taproot if its the Russian hybrid variety, is what you may want to get if you are doing outdoors and wanting to break up your clay.
As for mulch, you can just use aeration(s) if you have nothing established yet, which would be a good idea whether growing in your soil or not, to prevent air microbe and wind erosion. Then I would upgrade to stems, leaves, and other bio-debris, which is free, natural, and full of nutrition, but it can attract pests if you don't have chitinase-inducing ingredients in your regime.
Hay, meanwhile, is messy, I don't use it anymore.
Wood chips, hate em, they attract fungus gnats.. and dont interrupt any life / pupae cycles of pests.
Now I rock the DE Rocks (NOT powder, which clogs), topped with our glacial blend I call "CBD Sand", which is a major pest barrier that still breathes, supports air roots, and is fungal, great for flowering and packing on size without the worry of bugs. There are other options too, but this is a glimpse into our gardening life, in case it helps the cause!
Best,
Don