Steam mop or other tile floor cleaners

So my husband and I just bought our first house. The entire house has this beautiful grey wood tile with super white grout. It’s stunning, but it’s soooo hard to keep clean. My last home was laminate so I just used a regular mop or a wet jet to clean up messes. A regular mop and the wet jet do not cut it here, the grout stains so easily. When I move the wet jet over the tile, dirt gets trapped between the tile and it stains. I have attempted to scrub it up with tile and grout cleaners and a nylon brush, but it does nothing. I went out and purchased a Bissel tile steamer, but it also does not help the problem. If anything, it made it worse, because it deposited dirty water between the tiles and stained them even more (it’s a steamer so there’s no water chamber to empty out so idk why it did that). I’m at a loss, this is so frustrating and idk why the grout is staining so easily.

Pictures of the tile, the stained grout places are in the hallway and bathroom. I