If you do end up regrouting consider using an epoxy grout. I had worksurfaces I did that way in an earlier home, and nothing penetrated that grouting. It was a less-brilliant white than a water-based white grout, but it never discoloured at all. Sorry but I can't remember the brand name.

Mind you, it was a hard job getting the grout in, smoothed, and completely cleaned up before it hardened, which is a chemical reaction rather than a drying process. These days I would probably get a professional to tackle the job.