I also have rather stiff clay for soil. I dumped the roto tiller and “proper” composting several years back and replaced it with throwing stuff down and letting it rot. I combined that with lots of fall planted radish left to freeze and rot. I stopped freaking out over dandelions, dock and other deep-rooted weeds. I do use mulch of sorts, but it is just spent vegetable plants, pulled weeds and stuff I scrape up from the paths.
Used too I had about six inches, 15 centimeters of soil routinely ground up by the tiller on top of hard pack. Now I can work some of my soil with my fingers and pull up errant dandelions by the full root. Even nasty weeds such as perennial grasses and things that make runners underground are easy to control. The very qualities that made them so difficult to control work against them now. I just pull them up and instead of breaking and leaving a root to resprout, it just all comes out in one piece, and I just drop it back down, root end up. I don’t even care if it’s full of seeds.
I’m actually a bit shocked at the difference and in how fast it happened. Gardening is so much easier this way *(maybe not on a large scale) and everything grows wonderfully as a rule.
The only drawback is that some of my feral crops like radish, dill, marigolds have suffered a bit without any bare tilled ground to lay dormant in so I’m having to be more conscious of them and make sure they get dispersed in a spot they can colonize in fall or the next spring.