With stains and popping out filler-you got ground in dirt and old sealer that will prevent new sealer from working right. Sounds like you need a fresh start with this tile- This is how I refinish travertine-
Strip with stripper
scrub grout joints and tile surface with a stiff hand brush using lots of hot water
with a little dish soap in it to help degrease everything. Remove bulk of water by dragging a bath towel over it- ringing out collected water into a bucket- Then - while its still damp-
Wet sand whole floor with 400 grit wet sand paper wrapped around a firm square sponge. (sold in auto parts stores for wet sanding cars/painting prep) Comes in typing paper size sheets for about a dollar per sheet- a kitchen floor will use maybe 4 sheets. (Wear soft type knee pads or a folded towel or you'll get sore knees) Have a bucket of water with some dish soap in it- this time to make it slippery- sand one tile at a time- in a puddle of water- so it slides easily- press down evenly- but not firmly- paper should feel like its gliding on tile- sand in overlapping dinner plate size circles- go over each tile 3 times. Do like 4 tiles at a time- then mop up the dirty water with the bath towel. NOTE* first buy one sheet of 400 and one of 600 grit- go home and TEST clean trying with the 600 and the 400 in a hidden area. Once its dry- observe results. 400 can leave scratches if you press too hard or don't sand an area evenly enough- 600 may not do enough. You'll get a feel for it real quickly.
Rinse whole floor with clean water using a hand towel dripping with water- mop up water with same towel- dry whole floor with clean bath towel. Leave it for at least 4 hours to really dry out- a day is better- Fill all holes. Then seal with commercial grade sealer- one with a 10+ year guarantee on it.
You may want to add a commercial clear coat over it- knowing you want it to be easy to clean. Its made to coat every type of floor surface (except for polished marble and shiny glazed ceramic) Mop on 2 coats- $17 bucks a gallon- GREAT results. It's what they use on all the floors in shopping malls, resorts.
Or rent a floor polishing machine with the red pad on it- use with plenty water.
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I've been a tile contractor for 7 years- specializing in installing large travertine tile. I
also specialize in "fixing" bad travertine tile jobs- reducing lippage, refinishing surfaces, color correction. I've been a carved stone mural maker for 4 years- meaning: I have lots of wet sanding experience.
Good luck with whatever you choose to do!