For here in Kansas, the trick seems to be to plant the cover crop in the FALL, after the earliest vegetables have been harvested. I let the cover crop grow until frost killed it, and then in the spring I would mow down the dead, winter killed cover crop and till it under.
I have HEARD that some people us rye as a cover crop because winter will not kill it, but I always thought that any surviving rye could become weeds. I always wanted my cover crop to be DEAD by the next spring.
Austrian oats add more nitrogen to the soil, which was good, but the oats survived more frosts and so added more organic matter. Those oats were 12 inches tall before they finally died.
Both did good things for the soil, but I had to work for it because using a tiller takes effort. I have a very good tiller, too. It is fun to use when you are tilling a straight line, but it is a very heavy machine, and I feel that weight at the end of each row when it comes time to turn around and go the other way.
I no longer use the tiller for that reason, though last year my son tilled up a bit of the lawn for me so that we could have sweet corn.