Not my mother being against drilling any cabinets into the expensive bathroom tiles, then accusing us of "not being interested in helping her out"

You dont drill through the tile. You remove the tile and locate the studs through the drywall.

Cabinets are usually mounted to the walls, not the floor. You can place them on top of the tile floor, then level them with shims, and then screw them into the wall. The toe-kick trim, skin, or moulding will hide the shims.