Commercial contractor here….I’ve never seen paint applied to concrete….usually it’s some type of special coating like an epoxy which requires a high level of skill to look proper. The biggest issue is moisture vapor emission from the concrete itself that loosens the paint/coating. Companies who apply finishes to concrete will test for the slab for water vapor emission and if it exceeds a certain amount they apply a special sealer that typically costs as much as the finish product. Ground and polished concrete is a popular finish these days but it is an inexact science particularly when there are previous finishes as you are experiencing. You can buy less expensive curing/sealing compounds but these must be reapplied periodically to look nice. Even new slabs that are ground and polished may look bad because of different distribution of the aggregates within the concrete mix. This is not an indication of poor concrete - more a characteristic of concrete itself. It’s also difficult to grind concrete next to a wall…..you must use a different machine so you end up with variations between the ground finish on the concrete. We try to talk owners out of it, or at least convince them to use a tinted product….gray is a popular choice.

There are penetrating stains made for concrete…..typically you powderblast or shot blast the floor then acid etch and then apply the stain…. There are also concrete sealers…..as always the prep is the most important step. I highly recommend NOT using a clear one.
I would not use paint….

I think I would try contact cement on a couple of small wooden blocks within the field of 1 or 2 tiles (don’t overlap any grout joints) and then screw the holder to the blocks…