Even left-leaning economics like John Kenneth Galbraith have tended to oppose rent controls for the same reasons the Chicago School would despite frequently high support for rent control.

As for the “new construction raises prices” bit, here is my theory.

When a new large project gets built they run newspaper ads and billboards with eye popping prices that shock everybody, particularly small-time landlords that are like “These rates are 2x or 3x what we charge, we’d better raise the rent”.

There was a study in Colorado that came to the conclusion that it is impossible to build new affordable housing just considering the cost of materials and labor, not mind finance or land costs, overhead of permitting and all that. If we could go back in time and make ourselves more old buildings those might be affordable, but inflation and other social changes mean prices that support new construction will only be affordable to people in the top 20% or so.