None of the way housing is implemented is fair, it's all entirely unfair to everyone involved. What you are proposing here is even more unfair in that people who already had a house had to simply suck up not being able to get a house in Shirogane, because if an FC or person is legitimately interested enough in the Shirogane housing to completely give up their old plots and the fact that those plots are now on the market means nothing for that fact that you specifically want a Shirogane house as well.
No, let's not pile unfair on top of unfair. Even if people couldn't move their houses, the plots would have sold out just as fast. The moving feature had little to do with that because ultimately the total number of available houses went up across the board. The real problem is the limited number of houses with the way the wards are designed. If the wards expanded dynamically based upon how they fill up there would be no issue at all.