I have two clover grass plots, one has red clover (long stalks) and the other has white clover (short dense crowns). Both appeared naturally as a result of mowing longer weeds, are easy to mow, don't turn brown, are very drought resistant and bee friendly (bumble bees like the red, honey bees like the white). They are an excellent source of clippings for compost, survive mowing and are both easy to walk on, although you are aware that the long red stalks are being pushed over and down.
I think perhaps your lawns turn brown because you do not collect your clippings? I always bag mine as I mow to make compost. If the large trifoliate leaves are left on the ground they will of course dry and be much more obvious than thin grass leaves left on the surface. The collection does not harm the grass since the clover can make its own nitrogen and has a deep root system which survives a lot of abuse.
To solve the brown problem I think I would collect and compost the clippings, or try much finer mulching if possible (cut the clippings into much smaller pieces); and to solve the squashing sensation just increase the frequency of cut. You can oversow with finer grasses but I don't know that will help or be of benefit in solving the problem. Clover makes an excellent grass surface if you can put up with it being a bit rough. Clover works well in dry full sun areas, fine grasses are more suited to moister areas.