Sure it does. People aren't living in tents (generally speaking). Population can't grow unless the housing supply does.
There might still be a growing demand for Bay Area housing, but without supply growth, the effect of that demand is to push prices up until current residents are forced to leave. NIMBY housing policy has a lot of shitty side effects, but it does succeed at the basic goal of maintaining low population density.
The old Victorians you see on postcards were built as single-family houses. Most of them are now subdivided into multiple apartments. Many houses in the city have illegally built inlaw apartments in the basement. People live with multiple roommates into their 40s and 50s.
It is true that eventually housing prices become so high they outstrip the ability of the local economy to provide salaries. In the meantime, it massively increases the costs of doing business, the capital requirements necessary to operate any business in the city. So far, venture capitalists have been willing to provide that capital, but that can't last forever. Last time (1999), it ended in a big economic crash.
Further complicating the analysis is rent control: most existing residents aren't forced to leave, because they are legally shielded from rising property values and rents. They are given the reverse, an incentive to stay in place no matter what, because they enjoy an artificially low rent that they cannot possibly replicate in any comparable dwelling. This causes market rents to skyrocket as maintenance costs and profit are shifted by landlords to the few open units they have, subsidizing the long term residents. It's a wealth transfer from "people who have moved recently" to "people who have not moved recently."
Finally, there are tons of people living in tents in San Francisco, which has become a major political issue lately. This is arguably partly due to homeless people from outside the city moving in to take advantage of its extremely generous social welfare programs for the homeless, but it is at least in part due to high housing costs.
Population can grow even if the housing supply does not, if the existing population becomes concentrated into fewer dwellings.
Bay Area residents who used to have their own apartments or houses are now renting rooms and sharing apartments or houses with others.
The people moving here are generally wealthier or higher paid than the existing residents, and will not be deterred from moving here until the number of high-paying job openings decreases.
Regarding living in tents, there has been a noticeable increase in the homeless population in recent years. It's not clear if they were displaced from housing or came here from somewhere else, but there is no denying the increase.
* Allow new higher density development, accept that your neighborhood will be under construction for years and then will look vastly different forever; and besides that, the city has to divert budget from other things to pay for building expanded infrastructure now (because California property taxes are mostly fixed by Prop 13);
* Fight development, the city spends that money on your pet project instead, the neighborhood looks the same as it always has, and then there's the bonus that your house will be worth a whole lot more in 5 or 10 years.
This is why you have unremarkable 1950s-era 3 bedroom ranch houses that would go for 75k in most US suburbs selling for $2m in Silicon Valley. It's why you can pay $5000 a month for an apartment in San Francisco that would be $700 a month in most US cities.
Empirically, tons of people are moving to the Bay Area every year, whether local governments like it or not, whether they make provision for that or not. It's happening, and there's no legal way to stop it.
Resisting new development doesn't stop people from moving here, nor does it remove the need to build out new infrastructure. It simply makes property values become that much higher, because there's a fixed supply and a growing demand.
This is, I think the main factor in why NIMBYs resist development: their own property is worth much more every year. Building new residences would only slow the rate of appreciation of their property.