I absolutely agree that short lawns make no sense whatsoever. How long you allow the grass to grow, and how much of your lawn to allow to grow long is up to the gardener.
Once you start allowing grass to grow long and allowing it to flower, you become aware of the numerous different species of grass.
Ideally long grass ought to be cut once a year, in Autumn (haymaking), and the cut grass removed. Otherwise the long grass will die down naturally, and the dead grass will make it more difficult for the wildflowers that you already have to grow. Buttercups will survive anything, but clover seems to grow most vigourously in grass that is kept reasonably short (though not mowed when the clover is about to flower).
Removing as much cut grass as you can (in Autumn) depletes soil fertility and may enable more wild flowers to colonize the area.
We'd really like to see a snap if you have one.... when the sun comes out.