Fixing the "problem" of people not moving out when the area prices go up is exactly the problem itself.

The issue is availability of property flat out. As long as there's no incentive to make affordable property available, every actor wants their property values to rise and cares little for the negative effects it has on everything else.

Those property taxes exist for good reason because you have to be able to pay for the services / infrastructure from it. Without those matching the area, problematic spirals occur.