Yes, if they haven't yet turned it into bare dirt you should mow it when it gets tall and rank.
The birds are out there every day eating the plants they like while ignoring the ones they don't. This is automatically selecting in favor of the disliked plants to grow and grow as the desirable plants are eaten down. By mowing you are leveling the playing field so to speak to give the desirable plants more of a chance to continue growing while cutting back on the ones they don't like. At the same time the birds much prefer young, tender grass to old, coarse, tough grass so by mowing you eliminate the tough stuff and encourage the growth of new, tender grass.