Whatever groundcover you choose needs to be compatible with the requirements of the roses. Creeping thyme prefers things on the dry side, as does both the lavender and the rosemary (btw, creeping rosemary is not as winter hardy as upright - doubt it would survive that zone) and roses like frequent irrigation and fertilization. I'd consider a creeping form of hardy geranium or Veronica 'Georgia Blue'. I'd also intersperse other low growing perennials.......geum comes to mind but there are others. Perhaps dwarf penstemon.

While at Rainbow Gardens in San Antonio this morning I saw a whole table of plants in 3" pots that had the company name of "Stepables". There were lots of interesting plants for between stones and other places, plants that would take light foot traffic; various kinds of sedums, some tiny oxalis, and others that I didn't recognize, but looked like they would work. Nearby the nursery has placed dwarf mondo grass and frogfruit as mentioned above. Probably not all the company offers on their website would be suitable for Texas, but maybe one of your local nurseries would have a selection of "Stepables" that would work where you are.

Ditto Ken’s advice about not digging out the center. How long has your plant been in the ground? Have the remaining stems rooted? Did you shear the top after it bloomed? I’ve had a meandering patch going for probably ten years. Some years there are brown patches and they’ve always occurred after blooming. I’ve never been comfortable trimming out the dead stuff because the stems are so intertwined (and some stems only have foliage at the ends) that it’s hard to figure out what’s really dead. I’ve had good luck if I shear the dead blooms and the very top layer of the underlying foliage every year after the bloom dies. If the outer stems on your plant have rooted, you should be ok. It’s possible that rooted sections in the dead area will produce new growth, but I’ve actually only had that happen with a bushy thyme that also behaves like a groundcover thyme.

I planted “Annie Hall” creeping thyme in northeast Ohio & Kentucky and it bloomed fabulous the 1st year + every year after. Actually had some of this in a potted plant, put it in the ground in eastern TN & sane thing. Meanwhile I planted Pink Chintz 2 years ago in TN… healthy, spread nicely but very minimal flowering. SO disappointed… should have planted Annie Hall creeping thyme as it’s doing fabulous in eastern TN (knoxville / Chattanooga areas)