+1 If the tiles are 9" by 9", the are likely ACM (Asbestos Containing Materials). The tiles themselves are not the real issue because they are not very friable. The Mastic is the issue and caution should be taken when removing that as the previous poster mentioned. As a general rule, if the mastic is black, get it tested at a minimum.

Your palm sander is probably your best choice; a belt sander may be too aggressive. You are probably already sanding down to bare wood, although not sanding through the stained or aged top surface. A commercial two-part waterborne polyurethane will protect your floor and also help to 'glue together' loose fibers. Nothing will harden the wood at all. Professionals use the 3M epoxy adhesive floor repair 'gun' to treat ragged edges and slit grooves. We inject the epoxy under a loose problem area, such as a split groove or a flaking face sawn layer to support these problems and repair defects. The clear epoxy sets in about 5 minutes and cures rapidly after that. If you want no shine at all, consider using BonaKemi NAturale waterborne polyurethane...expensive stuff.

None of the DIY stuff hold up that well. Your garage mat can pull it off. Do a garage mat or paint. Not both.

Yes, it’s a garage floor. And it has not received a single blemish, chip, stain, peel, or scratch in five years. There’s a reason as to why that is. I paid a lot of money to have an Epoxy coating on it for protection. Scratches are very, very uncommon. The guy was actually surprised that it happened at all. Obviously, not surprised enough to fix it for free - but surprised nonetheless. I consulted with a few companies and they all said the same thing. Full sand down and roll a fresh clear coat. Most people would just deal with it - but my OCD says no. The floor installers also managed to scratch two ”scratch resistant” planks of the laminate flooring they installed by also dragging equipment. But thankfully, they will replace those for free.