Hello and thank you for asking about a lawn alternative. Leptinella gruveri (Miniature Brass Buttons) is best suited for sun and part-shade situations where it can receive adequate moisture. I don’t find listings for it being drought-tolerant. The short answer to your question is “I don’t know” if this plant will work well for your large area in place of turf.
I don’t have a single research-based answer from the OSU Extension Service which I can refer you to. This is still a field people are learning about, experimenting with, and studying. You’ll find different suggestions for lawn alternatives when searching the web and if you talk to your local nurseries. There are some reasons turf is still used. It can be seeded, so the cost for a large area is lower than buying pots or plugs from a nursery.  Herbicides for broadleaved weeds can be used when they do appear in turf, while “chemical weeding” isn’t really possible with a broadleaved groundcover. Turf is fairly sturdy for foot, dog, and chicken-traffic.
Strategies around these downsides of groundcover include using stones or pavers with groundcovers between them. Or, using areas with shrubs and mulch or bark to make the groundcover area smaller for easier maintenance. I have also seen different groundcovers used, creating a sort of tapestry, which allows plants to be in their best zone, such as one for shade, one sun, other types in dry or wet parts of the property. Whichever choice you make, September is generally a good time to get plants started. The soil is still warm for root-growth, and the rains will soon begin so your watering period is limited.