What I have done is planted a cover crop that winter kills. Then it covers and protects the soil for the winter, and you plant in it in spring. Only problem is if you have a surprise mild winter, then it doesn't kill (I've used oats for that purpose.)

Interesting question -- will be interested in the suggestions you gather.

I have used a sickle mower to get the kind of effect you are describing. I use a Troy Bilt mower. The mower tends to cut the plants at several places as it passes, and the blade pushes them against the soil, so the pieces do not tend to blow in the wind. I've done this very successfully with buckwheat and wheat. Clover was not as successful: lots of the plants recovered, as might be expected for a perennial such as clover. Oats tend to make a mess in the mower if left too long; I would avoid them. Corn actually is a good candidate for this kind of treatment, and you can pick the ears before mowing. I find the corn stalks are really a pretty good mulch, and do not decompose too quickly.

Renais

You need something that will bend the grain stalks over and cause a break to occur in the stem so the upper part of the plant dies. A heavy bar might do that in a garden and walking on the planting bed will do that (I have done that)
As Peter pointed out the crop must be an annual, and clover can be a perennial, so it will not work with crimping the growing cover crop.
Mowing the cover crop is not the same thing as crimping either.

On a small scale one could probably trample, as Kimm says, or maybe beat it with a flail of some kind.

I've been thinking about the crimping lately, too. I have a two-wheel tractor and someone has begun making a crimper attachment for it.

What would a lawn roller do?

I like tillage radishes.

That may be a use for that implement today.
Radishes will open up compacted soils for a short time but if the amount of organic matter is not increased to optimum levels as well the soil will revert to being compacted again.