There is some possibility of confusion about what "steady state solution" means here. I would take it NOT as meaning uniform temperature everywhere and no heat flow -any form of insulation would delay reaching that state, but would not change the damage caused when it was reached. Rather, I would take it as meaning a uniform heat flux through the thickness of the tile, and therefore a linear temperature gradient through the tile.
Q_Goest has given the most detailed analysis on this question, which I believe is in good agreement with all of your points, except for the significance of the heat capacity of the tile. I found it very persuasive in spite of my earlier concerns on that exact point.