Mmmh…during the design period we were also discussing about scaling the difficulty by introducing obstacles, in order to potentially restrict the number of available tiles for each move and raise the chance that you would need to reshuffle and lose time. This could lead to situations where the player has to do it too many times in a row and feel frustrated about the RNG, tho.
I guess your solution could also work, but adding a cooldown could ultimately put the player in a spot where they feel restrained: tiles get chosen completely at random, so you could get stuck and lose. There would need to be some safety net (like assuring the presence of at least one placeable tile in some way?).
Again, thank you for your feedback, we will surely take it in account :D