Except I said that relocating would put you in the lottery too, so I've covered that issue. You cannot trade houses anymore. Relocation of any sort, or buying a house for the first time, will now put you into a lottery so that everyone is on equal grounds. House selling is much harder now (Shell FC that owned a house previously can do it still to some extent).
Optimally no system is needed though because everyone has a house.
You describe a supply(/demand) issue, I suggest we fix that by making a larger supply (to the level that every single person and FC with a desire to be in on the system can be in on it). Your suggestion is to fix supply by lowering demand (making them more expensive so less people want them). Both address a supply issue - yes, but only one of them gives people content that they wanted to take part in. Perhaps short term we could use your solution, but I don't think it's good for the game long term and will cause a bit of a kerfuffle (so if the solution happened without news of a fix to come later, pretty sure it'd just make people upset lol).
The end solution will always be to increase supply (if one is interested giving people content at least). Besides this isn't some hardcore Mortal Online mmo this is a theme park, so it would fit the theme best if every single person and FC that wanted a house could get one. In fact I'd argue the best solution would be to give every single person (and FC) a small plot of land by default (it just happens) and then add significant amounts of build up in that (such that it can be a large gil sink but the barrier to entry is near nil).
Getting people in on the experience and encouraging them to make a home I believe would be better for the game and the system itself (as there is justification to improve housing more then). Beyond that when you wet more toes more people might be willing to go into the deep end (improving markets related to housing both in game and cash shop). Also not threatening to remove (demolition) it if they don't show up in a while would be a nice encouragement to them being interested in it lol (but that system has to be as it is until supply is fixed).
Going back to that everyone starts with a free mog room experience. I strongly believe that housing is a brilliant idea for mmos, a home within your home away from home, and an MMO should prioritize creating that experience for players earlier than later (at least one that isn't hardcore and does things like drop loot on death / real world space no instances).
So in other words while your solution is lower effort absolutely I don't think it solves the issue in the best way. Getting everyone a home would be better, imo (but obviously more challenging task for SE to figure out- although we know it can be done in a general sense as we've seen other games do it).