I'd diamond grind and put down ceramic tiles, or epoxy, or Terazzo.
Staining is best done on a brand new slab with no big divots or saw-cut and replaced patches - because the concrete mixes are always different, meaning the stain never sinks in the same and it'll be glaringly obvious. For an old floor like that, you may be better with a solid color two-part epoxy.
The most permanent and aesthetically pleasing way would be to clean it all up with a diamond floor grinder to get down to clean solid concrete and level, and don't worry about the little divots. Then thinset mortar and ceramic tile. It's a HOUSE, you want to use treatments normally found in a house. Stained concrete usually isn't, unless you're repurposing an old Brewery Building or something.
Tile is usually considered expensive because it's labor intensive - if you buy your tile at the right price and put it in yourself (with friends) to save labor, it's not that much more than staining and sealing. Just get a few boxes of spacer crosses and start from a common baseline and work out.
Save a bunch of extra tiles, and they can be replaced if you ever break any. And if you get Mixed Dye Lots or Production Runs to save money, be sure to either randomly shuffle them to spread the shade variations, or use different lots in different rooms.
Note this won't be a good idea if this is not slab-on-grade, or a nice thick structural slab over a basement. A 2" pour of lightweight concrete over a wood framed floor (soundproofing) is NOT a good candidate, it's going to move and flex too much and the tiles will pop loose.
They do make special thin-mix self-leveling concrete mixes specifically for leveling floors like that, they will do pours anywhere from 1/4" to 1" thick for when the old floor slab looks like 25 miles of bad road. But it isn't cheap. You do the diamond prep to get all the paint and loose stuff off the old slab and get some "tooth" for the new poured layer to stick.
Another option would be Terazzo - Marble Chips in colored cement - Hollywood Walk Of Fame and lots of Supermarkets from the 1940s era. Pour and cure, then diamond grind to polish the marble chips, and apply floor wax. But it'll need to be a good inch thick.
And it can be patched almost seamlessly if you keep extra matching color marble-chip mix and can recreate the exact color formula for the cement - I can find the patches, but I have to point them out to others.