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Since you really have to use physical abrasion I'd get yourself a wire brush drill attachment for an electric drill and use that.

Easy. Test light solvents like mineral spirits on a corner spot or extra tile.

Yeah I'd worry about chemical seeping in there and permanently discoloring the tiles. I think physical removal is your only answer and frankly I'd suggest something like a sandblaster. Unless the top of the tile is coated with something that you're trying to preserve the terracotta is going to be terracotta all the way down so you just got to physically remove the material that has the paint and I imagine the paint has soaked into the porous terracotta tile as well.

jfc
You boil a pot of soapy water and the dump it on the spots and scrub. It should come right off. Do you know anything? Or do you always use hired help?

paint thinner is the only thing that's going to possibly work.

I know, but refuse to say. KYS.