I think if I was going to have an edible lawn, I'd want one that doesn't need mowing!
I have planted some cammasia, but they are doing better for me where they don't have the grass competition, although those are a different variety so not a good comparison perhaps. They've been in a few years now and the ones in the grass have almost stopped flowering. I think it's probably time to dig and divide them and see what they might taste like!
My thoughts for an edible low maintenance lawn would be to try and replicate the silverweed/clover gardens of the Pacific NorthWest. I have a few strains of silverweed I have collected and it does grow well here, even in grass competition.
Silverweed - these roots are quite knobbly but long - usually they are smoother.
I really like the taste and it is apparently higher in starch than potato by weight. I'm thnking that the clover would help maintain the fertility and the roots of the silverweed would grow bigger as a result. I've been a bit inspired by the links in
this UK blog I need to find out a bit more about Springbank clover (Trifolium wormskioldii). If it likes the same conditions as silverweed then it should do well here, but I haven't found any information about it's palatability and practability as an edible.