[dice]

I think a mulch mower would be good, if you do not have to do
too large an area. When I planted winter rye, I decided it was too
much of a pain to turn the top growth all in. So I mowed it with a
hedge trimmer (weedeater would work, too), raked the top growth
off to the side, spread some compost over the stubble, and turned
that over with a shovel. Then I raked the top growth back over
the bed for summer mulch.

(By the time it flowers in late April, winter rye is typically chest-high,
so it needs to be cut down before trying to mulch it up finer. I tried
putting it through a shredder green one time, it bound up the shredder
like rope, and I had to disassemble it to get it all out of there. That
might work if you let it dry out first. No-till farmers knock it down with
a flail mower and just leave the top growth on top of the soil.)

Not all cover crops have top growth that challenging to amend into
the soil. Vetches and clovers are fairly easy to amend the soil with,
and I expect that a mulching mower would have no problems with
them.