Clean porcelain stoneware with an eccentric grinder?

Good day! I have porcelain stoneware flooring throughout the entire house, and in the more heavily used rooms, a certain dark layer has built up over the years, which cannot just be wiped away easily, but it does come off when you press hard with a sponge and a white natural cleaning powder. Of course, this cannot be done on large areas. Therefore, I am considering whether it might be a solution to use an orbital sander for this. I have two polishing sponges for that and would take the somewhat harder one. But what product should one use for this? It should not be too abrasive... Matthias thanks you!