Michael Bell writes

On a related matter, I am not sure how to handle the beds around these
lawns, or even whether to have them at all. I have never seen the bare
earth of flower beds as beautiful.

Flower beds don't have to have bare earth, and they're a lot easier to
maintain if they don't. A dense planting of small perennials (not
herbaceous plants which die down in winter) keeps the weeds down, and if
you choose plants which are interesting for their leaves, bark or
berries, you can get a garden which is interesting all the year round.

On the other hand, grass is the perfect ground cover, (it grows
anyway!) and I could sow the lawn to the concrete edges and either let
the flowers grow though it, or cut little holes for each plant, and
fill them in again after. But that creates difficulties with mowing
the lawn.

You could try a wild flower lawn - choose low growing things like
daisies and clover and self heal and birds-foot trefoil

You need to work out why the current lawn isn't doing very well - poor
drainage, shade, trees taking moisture away? The same conditions will
affect anything you plant there, so you need to adapt your planting to
the conditions you have.
--
Kay