Earlier quoted context omitted.
So if your neighborhood gets popular you get kicked out? Sounds like a healthy policy. Have you ever thought more than 2 minutes about it?
You could provide exceptions for a primary residence where the increased taxes would be sufficiently burdensome on the owner's income. Doesn't have to be all or nothing. Just don't do what Prop 13 did in California where they applied the property tax limit even to commercial property for some reason.
The California approach is to let the owner get massive appreciation, which they receive at sale, but never pay tax on any of it.