Mowing high is not an elimination technique. It's a coexistence technique, with benefits.
The grass turf will keep growing and crowding out weeds. The low clover will crowd the grass for sunlight, retain moisure, build the soil, and feed pollinators (and rabbits) through the whole growing season. I think this is a perfectly satisfactory arrangement.
And it cons the neighbours into thinking you have something like a lawn! Silly neighbours.
