Earlier quoted context omitted.
What if the city lets you declare your chosen value without being able to force you to sell, but if you ever sell at a higher value then you owe back-taxes on the difference? And if the difference is more than X% then it’s fraud unless you can persuade a judge otherwise. The loophole might be that Billionscorp LLC is listed as the property owner, and Jeff Billions technically only rents the penthouse from his own com…
You'd just end up arguing about when the property appreciated. The owner would say it all happened since the last tax payment was due. It would also complicate the home buying negotiation. People would look at your recent tax payments and put a cap on the bids they would make based on what would trigger back taxes for you.
You are right to imply that it seems unfair if you discover in year 49 of your happy 50 year tenure that your Queens bungalow was built on top of a seam of pure gold nuggets all along.