I don't know of anything that removes tile grout that doesn't hurt the pipes too. I have no idea why tile setters dump bits of tile, grout and mortar down plumbing drains. That is totally unacceptable building practice, and the tile setter will need to fix this. You had a perfectly working plumbing system, until the tile setter got lazy and decided to cement the drain shut.
All he had to do, was walk outside, and clean his tools there.
I had one job where the tile setter had dumped broken tiles down the toilet drain. I wondered if that was sabotage to the homeowner or just old fashioned ignorance.