One thing you may not have realized that by plowing, the soil is
opened and turned enough so that weed seeds lying dormant (for
30+ years and more) suddenly see the light and spring for it.
occasionally I find I have to disc or plow a piece of ground that has
gotten 'hidebound' but usually, I try to avoid making any opening
cuts in it.
Or, as in Germany, they have realized that if they can cultivate
(plow, etc.) at night, and mulch or cover over the new dirt before
sun up, the weed seeds never germinate.
Mowing, on the other hand, will cut the top growth down, using it to
mulch the soil while the root system is left in place to act as a
barrier against erosion, and to help rot happen below ground.
Usually if you have grass and weeds and continue to just mow
them down, you'll have more grass and the weeds can't get a