All of my grass is native, not a cultivated lawn. I also have a lot of acreage, and a cover crop would be prohibitively expensive to prep and seed.
Even if you can establish a perennial crop, the native grass can still creep back in and outcompete it. And if it can keep the grass down, the cover crop necessarily will grow enough to create the aforementioned problems with pests. You still gotta mow it periodically to keep everything under control.