Tile will stick to most rigid level surfaces including painted wood provided the correct preparation work is done to ensure the adhesive or mortar will stick to the floor surface.
Thinset over painted floor.
Cracks and gaps can be filled with portland cement based fillers.
Not that they would endorse going over an oil based paint what you should do is scarify the paint off the floor.
The polymer modified thin set mortars might bond to a latex paint but not an oil based paint.
Mapei ultrafor plus data sheet says to use their keraply latex admix in lieu of water when going over impervious surfaces.
Installing tile over painted concrete adds a few extra steps to your tiling job.
Unfortunately the tile cannot be adhered directly to the painted surface.
Mortar cannot be installed directly over painted concrete itself.
Tile cannot be attached directly to painted concrete as the thin set will not adhere well to the paint.
Whether you have cracks chips or just a stained concrete slab thinset helps cover the old surface and create a new problem free area.
The safest way to approach questionable concrete is not to cover it over with cbus but to fix the concrete.
How to use thinset morter to resurface concrete.
It must be easy to work with and it must adequately fill in gaps between the tile and the substrate.
Paint used on concrete will not bond with the tile adhesive.
All paint will need to be removed before you begin installing the tile.
It cannot cure too fast.
The house was built in the late fifties and the slab floor has been painted at least once.
Tile adhesive is expected to hold the tile in place not just for years but for decades without fail.
Otherwise you do.
This surface is called a scratch coat.
A scratch coat will give the mortar something to hold on to instead of the layer of paint over concrete.