Hi, you can get professionals to clean and seal your stone. You can also do it yourself, but I'd presume the professional clean would be better. I've sealed my own stone tiles and it was very easy. A simple wipe on then off process. From what I understand your stone is uneven surface. From that I think you can't put flooring on top without leveling it out. And considering it doesn't cover all the floor, I'd remove it if you were planning a new floor finish. But if your not fussed for the floor, and watching your budget, and plan to replace it at some stage, I'd get my DIY on with some stone cleaner and sealer.

I used a mix of baking soda and white cane vinegar and a lot of elbow grease.

Elbow grease 😊 E cloths are quite good.

I do this as part of my business. Those tiles will need chemically stripped to remove old "restore" style products, polishes, etc. Then the tiles need deep cleaned with a high PH cleaner, rinsed and left to dry for around a week. Then they can be sealed with an impregnating colour enhancing sealant, such as Dry Treat CE. This is a breathable product. These floors were normally matte unless they had an early powder glaze.