I too use leaves as mulch around set plants. Just leave them lay over the winter, and plow down in the Spring. They will pretty well be broken down into compost by then, and less chance to deplete N out of the soil. Night Crawlers and red worms love them, and will feed on them year round, if thick enough.2 pluses from that, is worm castings, and their tunnels, allowing water to drain away naturally.
I plant Oats on my potato patch yearly as a cover crop, but, I'm far enough North, I get a Winter kill. And, snow pack normally presses it right to the ground, so turning under is not a problem. Last year, I added Buckwheat which may work better for you. Adds lots of OM, and breaks down quick. Don't know that I will this year, because of rising seed prices.
Daikon Radishes make a great natural aerator, and see is relatively cheap. Roots can grow as deep as 3', and will penetrate hardpan. A good maintenance plan, once you get the hard pan broken mechanically. I sow some in row, as trap plants for cucurbits, and get the full benefit of the summer growth. I plan to sow some in with the Oats this Fall after I did potatoes. They will grow fast enough, to grow though any hardpan I may have in that area.
I also use a David Bradley walk behind tractor to cultivate my row crops, not use a walk behind tiller, at least most of the time. The only time I use my tiller, is when plants get too tall, or rows grow in, and get tooo narrow to use the D-B. But, I made a tail wheel for my tiller, and only till the top 2" of soil. Just deep enough to grubout most weeds, and leave them on top to dry and die. Any deeper, you're only bringing more weed seeds to the surface to germinate and grow. I originally built it to till through the leaf mulch, when I'd get weeds growing on top of the leaf mulch in wet years. Weed seeds came from some weeds that grew alongside the leaf pile, and deposited seeds in the leaves. Just wasn't proactive in keeping them mowed off, or sprayed. But, I've found the tail wheel works great for shallow tilling on bare dirt.