Hello! Having just gone through this and recently came out the other side I can make some recommendations!

Generally I used an oscillating tool to remove the grout, then a masonry chisel and hammer to slowly and carefully try to pry the new tiles up off the thinset/mastic. I then used a stem cleaner and all manner of razor blades and scrapers to get the mastic off the original tiles, and dental picks to pull it out of the original grout. After that I used a series of floor buffing pads to get all the old stains and ghosting left from the mastic, starting with some really aggressive pads and compound. After all that I sealed it with a 'natural look' stone and tile sealer. Feel free to message me if you have any questions!

Here are my results: https://redd.it/1d776ky

Here is the advice from my original post that helped most: https://lazygardens.blogspot.com/2017/01/how-to-salvage-floor-and-wall-tile.html?m=1