Not really. People would be jumping all over building new housing to solve their housing woes if that were the problem. That isn't really happening because new houses cost even more!
Realistically, the root issue is prolonged high employment coupled with our failure to bring much automation to home building. If people were desperate to accept any work that they can find, the cost of construction would be decimated, but so long as construction remains relatively unattractive work to a workforce full of choice it needs to pay a lot to convince anyone to do it; a cost that gets passed on to home buyers, and eventually renters by association.