I’ve certainly had cow-orkers that were just as useful as the office’s decorative plants. And after a few weeks, just as invisible.
And they’ll be lower to the ground than they would be otherwise: something you’re unlikely to notice unless you ordinarily see dandelions growing where they don’t get mowed. They do get taller.
A number of plants will set seed at a relatively tall height if left to grow undisturbed, but if they get mowed will try again at a lower height. I’ve seen redroot amaranth setting seed almost at ground level, where it’s been mowed.
I doubt this is a response specifically to human mowing, however; my guess is that it’s originally a response to grazing.