1) People not moving to markets like SF because rent is too high and telling employers that is exactly what should happen! This forces employers to either lobby for more housing (a good thing) or open offices in reasonable cost of living locations (also a good thing) instead of continuing the status quo bubble. Rent control is a market distorting force in this case.
2) Rent control directly leads to fewer new housing units because there is a disincentive to create housing that's going to rent for below-market. If you want more housing, you should be opposed to rent control. I say this as someone who benefits greatly from rent control and still wants to see it die.
3) The population of SF has not remained extremely stable since the 90s. Look at [1]. There were 140,000 fewer people in the 90s than there are today (a 20% increase)! Also your reasoning that "there's no more space for people to live" is demonstrably false. Go visit Tokyo and then tell me there's no space for more people in SF. There's no more space because zoning laws and height restrictions in the city are absolutely ridiculous and serve NIMBY interests.
[1] https://en.wikipedia.org/wiki/Demographics_of_San_Francisco
Edit: since someone is no doubt going to ask, here's a good summary of the economic literature on the topic. It covers a variety of effects related to rent control: https://econjwatch.org/File+download/238/2009-01-jenkins-rea...