A big patch of Common Blue Violet (Viola sororia) brightens my backyard. The white flowers sprinkled throughout are a form of this species known as the Confederate Violet (V. sororia forma priceana). I have encouraged violet proliferation by just not mowing them off while in flower, so they can set fruit. It's worked well and I've got scores of purple jots brightening the turf grass that remains. How this native violet could be considered a "weed" is beyond me (although a "weed" is in the eye of the beholder, I suppose). Moreover, spraying toxic chemicals to destroy them, in favor of a flawless emerald blanket of nonnative turf grass is crazy, in my opinion.
I made this shot of this Northern Flicker (Colaptes auratus) last April in one of my violet patches. I've seen these woodpeckers in the violets a few times, and I suspect they are hunting ants. Ants are major dispersers of violet seeds, and ants are a major part of a flicker's diet. If having a "weedy" violet-filled lawn means flickers, I'll take it.While the sedge lawn may, to some eyes, look a bit coarser than the manicured nonnative turf grass that we've been programmed to plant, manage, and cherish, I think the sedge lawn is a vast improvement. Another benefit is that invertebrate life forms have spiked tremendously. The sedge forms a duff layer that seems to be far more conducive to animal life than that of sterile turf grass. One barometer of success is the sheer number of fireflies displaying in summer. Last year they had expanded to the point of dazzling. It was like a laser light show back there. Viewing them from a second-story window reveals that the displaying beetles stay almost entirely over my backyard, with little drift into the neighbors' yards (most of them manage for turf monocultures). Firefly larvae are ground dwellers and predatory, feeding on various small animal life. I would say that the sedge supports vastly more of a duff layer food web, thus the huge spike in fireflies.
If you'd like to shrink your lawn, experimenting with Carex blanda might be worth a try. I haven't researched where one might obtain it, but I think some native plant nurseries carry it. Try googling "Carex blanda nursery" or something like that, and you should find some sources.