This is actually a key problem with using rent control as a primary means to achieve housing affordability. Rent control works as a way to soften the blow of large rent increases (like the kind you might see after a major building renovation), but it really only works as long as there are plenty of other units on the market so that when eviction inevitably does happen, the tenant has somewhere else to go that they can afford.

The problem is, many cities like NYC and San Francisco are trying to have their cake and eat it to. Liberal residents want to keep their suburban-style single family neighborhoods while simultaneously appearing to care about skyrocketing rents. You simply cannot have it both ways. The ONLY viable solution is to relax the rules blocking the construction of new housing. SB827 in California, which would have overridden local zoning to legalize the construction of higher density buildings within walking distance of transit stops, was a valiant effort to do the right thing, but it failed because of a strong anti-development sentiment from anti-capitalist tenants rights groups and pro-housing-appreciation single family home owners (in a particularly absurd moment, the mayor of ultra-tony Beverly Hills, which has built almost no new housing in the last few years, claimed to be against the bill because he was concerned about all of the "luxury housing" it would build in his town).