In theory, yes, but in practice what happens is those people get priced out of their homes anyway.
If a retired person was living in the $1.8m home in my example above and was paying a low tax based on the 1970's of the house, they might not be able to afford moving to a more practical $900k one-bedroom apartment/condo because their tax rate would increase and they may not have the cashflow to pay for that bill (even considering the proceeds from the sale of the home).
If you walk through SF, even really nice neighborhoods like St. Franciswood, you'll see large beautiful homes that are in severe states of disrepair because the owners don't have the cashflow to pay for the enormous costs of repairing their homes. They'd be better off moving to a smaller dwelling, but they won't because it could cost more and the incentives are more out of whack.