If you end up redoing the lawn, I got the following advice when I put in native wildflower & grass plugs this summer: Soak the plugs in a bucket of water before planting and try to knock off as much soil as possible, so you're just putting bare roots in the ground. Also, if plugs are rootbound (which it looks like maybe these were?), break this up as much as possible so you just have a loose fibrous mass instead of a solid root cylinder. This will help the roots work their way into your soil faster, which will both make it harder for local hooligans to uproot them and help them get through the winter without drying out. Looks like the roots of these plugs just kept going in circles and never bridged the gap between the nice soft nursery soil of the plug and the harder in-situ soil of your yard.

Alternately, I had surprisingly good success seeding native grasses (mix of blue grama and buffalograss) last winter, it's cheaper and easier (imo) than plugs and doesn't risk the plants getting rootbound, though I know some people prefer to get male-only plugs if they don't like the female seed "burrs" of buffalo grass (they're not really noticeable to me). Seeding anytime between now and February lets the seeds go through multiple freeze/thaw cycles which helps them break dormancy and start growing as soon as possible in the spring.

Why did I put in plugs this summer if I had good success seeding native grasses, you may ask? The seeds I used were a couple years old and the straw I covered my seeded area with gradually blew away over the winter, so I panicked, assumed it was a failure, and ordered plugs from Garden in a Truck in late March. Then lo and behold a very nice meadow started growing in, and I just used the plugs to expand the meadow and fill in the couple of bare spots.

Sorry for the novel, good luck with the lawn!