Suppose you get a job offer, or your adult child has a newborn, or your aging parent's health takes a turn for the worse. Instead of being able to simply move, this instead starts a decade-scale process that has a 1% of chance of allocating you a public housing unit in the end. What does that do for dynamism?

People's intentions about when and where to move are important. A housing system that removes all individual agency from this question, abdicating everything to a government lottery/waiting list system, can meet other desiderata but is clearly losing something important.