Certainly a very hard inflexible rent control does contribute to local housing shortages, and can also cause housing stock quality to decline more quickly, but more moderate rent control regimes, where annual rent increases are capped at a reasonable level, can insulate renters from short term sharp rent upswings and abusive landlord practices without really having a big impact on the housing stock.
There are a couple of issues here. The first is that housing supply is short run highly inelastic. You can't build new units overnight. The second is that while rent control can cause housing shortages in the technical sense - that at rent control rates, more people want those units than there are units available - what causes housing shortages in a more colloquial sense is zoning laws which restrict new housing construction and drive up prices/rents in highly in demand areas. You know, nobody wants to live in San Francisco because it's too expensive.
Anyway, the point is that people who think rent control, which while potentially flawed does provide benefits to renters, is the devil tend not to think the same of density restrictions and limits on residential construction.
density restrictions + rent control = lower rents, more people want to live there at those prices.
density restrictions without rent control = higher rents, people would like to live there if only they could afford to.
Neither regime leads to more houses being built, but one leads to it being more affordable.
