Ideally, if one company charged too much for rent, the person living there could move to another place owned by a different company that would charge cheaper rent, in an effort to take business from the first company.

The issue arrives when monopolies control the majority of homes, but those monopolies can be broken much more easily if the cost of buying homes went down (from an increased supply of them) and the barrier to entry of new homeowners who want to rent went down. That would increase competition and would eliminate these monopolies.