Without rent control, the market is free to compete and suddenly open up a ton of bottled up apartments. The issue is, a family on welfare with a income of 40k a year should not be living in a 4 bedroom apartment in pacific heights for 1k a month. I have no problem with giving affordable housing to those in need, but rent control is not the answer as it places the burden on individual landlords to provide social security.
The biggest gap of people who can live in SF are not the poor (taken care of by rent control) or the rich (dont care/buy homes) but those in between. Run of the road engineers and others who make between 75k and 150k a year. These are the people who are hurt most by san franciscos crazy rent laws.
If rent control was abolished, the one thing you would see is that rental rates would lower significantly as supply starts to balance with demand. Yes, some people would be displaced but the question is- If the city of san francisco wants to provide affordable rental units, why dont they instead just pay lower income people directly, letting them be free to choose where to live?