Interesting idea, but there is the point that people would be able to see the empty plot, unless the old house doesn't get demolished until the timer is up.

An idea I'd like to see is where you put down money for the size plot you want in the housing district of your choice in a queued system, then when it's your turn, depending on the price of the plot (You would put down the maximum value of that plot size; ex. 3.6 mil for a small) That money you put down would be put toward the plot and if there is any gil left you get it back. It wouldn't be a quick solution but it also can somewhat guarantee you getting a house as long as you're still playing the game.