For a few reasons that I won't get into, I grew up around Portuguese bullfights, which are fought on horseback. The bullfighting arenas are empty most of the year, and they actually naturally grow grass in the pit, but they usually clear the grass with tillers before bullfighting season starts. I've seen people ride in grassy pits before, practicing their bullfighting maneuvers, and it is actually pretty amazing how quickly the grass turns to mud. The key thing is the type of maneuvers they do. They start and stop alot, but they also do this thing in the middle of a charge where the horse alters its momentum sideways using its hind legs. After about 20 runs or so, the outer part of the ring is muddied, and after about 50 runs, the inside of the ring is muddied.

Also of note is soccer fields. Note where the most worn areas of the field are: the two areas where the goalies stand. The thing that the goalies do on the grass that the rest of the players do not: they jump, and most often laterally. Again, movements that change direction quite quickly.

I can understand why grass doesn't turn muddy on modern battlefields where guns are the primary weapons. But I wouldn't doubt for a second that a medieval battlefield would be a muddy mess. It may take a long time to get rid of the grass by marching or walking, but try fighting on it with horses while ground forces collectively stampede each other. Those aren't normal movements on the grass...they're going to dig in and uproot grass quite easily. And since guns aren't involved, physical proximity is closer, and therefore the impacts are more localized and intensified.