@rabbitseatdogs have a look at la redoute for comfy and nice linen dresses. I bought a few last year from them, midi length, with pockets, linen or linen mix, in black, mid grey and khaki. They had other colours but I don't really do other colours. They have something along those lines every summer.
School run dress is just mn shorthand for comfy, practical summer dress. Not necessarily for actual school runs. Daft thing to be getting all aerated about.