Agricultural extension research indicates that it can be cut shorter than the 3 inch recommended height. Read the PDF for more details. In my polyculture field there are some patches which I try to let go to seed and try to cut the grass seed stalks above it but sometimes have to cut close to the ground to deter unwanted plants. It comes back up vigorously shading the soil so that it dominates that spot.
alfalfa yields are better cut lower at around 2 inch height, grasses no lower then 4 inches. Cut alfalfa on average every 28 days and no later in the growing season then will allow regrowth for winter survival. In my area I usually cut no later then September 15.