Tax caps are the lesser evil. Without them, time and time again, city management has shown irresponsibility.
The proposed modification, to a per-person cap, solves your complaint about new people moving to the area. If the cap is $4,000 per person and there are 40,000 people, the budget is $160,000,000. If another 10,000 people show up, for a total of 50,000, then the budget rises to $200,000,000. Each landowner pays their portion of the resulting budget according to land value.
The proposed modification, to a per-person cap, solves your complaint about new people moving to the area. If the cap is $4,000 per person and there are 40,000 people, the budget is $160,000,000. If another 10,000 people show up, for a total of 50,000, then the budget rises to $200,000,000. Each landowner pays their portion of the resulting budget according to land value.