I can’t speak for every local govt but it typically doesn’t work like that. Taxes are based on a millage rate. For example, my county did a reassessment of all the houses in the county. Home values have ~doubled since the last assessment but our taxes stayed roughly the same because they just change the millage rate.

If local govts got more money when real estate prices rise, they’d be swimming in money. But they propose and pass a budget. The only reason your taxes will rise is because their budget became bigger, not because real estate values changed