Lawn flowers & biodiversity sugestions please
 

My front lawn suffered from the wet summer & there was a lot of moss in the shade + a build up of 'straw matting' in sunny areas.
I've scarified & lifted it. There is need to do replanting in bare & balding patches.
I'm not interested in a green desert & much prefer a carpeting of small flowers (daisies, white clover & forget-me-nots are well established).
Can anybody suggest additions along these lines to go with the grass seed?
I divet out broadleaves like dandylion & I'm not looking to encourage anything much bigger then daisies.
I'm also wary of introducing anything which might get totally out of control & invade the surrounding flowerbeds.
We pay somebody to mow & strim the lawn every couple of weeks during the summer but they don't cut it very close.
Surfing the net most stuff written is either by show gardeners who seem to regard everything other than cultured grass as a pernicous weed & enviromentalists trying to encourage meadowland a foot high.
It's in the SE of the UK. The soil is fairly light & anything acid loving tends to curls up & die.