It seems to me the two factors most responsible for a shortage of housing must be (1) increasing inequality of wealth and (2) legal restrictions setting a lower boundary on the kind of housing which can be provided.
It is not the case that the real-estate market is rigidly inelastic. Housing units can be resized so that there are more or few in a given land area, and land and buildings can be redeveloped so that their use changes. However, such changes do require significant investment, which bears on the issue of whether rent control affects housing supply.
If everyone in a real-estate market has about the same wealth, the best strategy for each will be to hang tough and try to buy cheap. Landlords will recognize this and a balance will be reached between their desire to increase revenues and the tenants' desire to reduce payments. The result ought to be a large, stable number of small, cheap apartments. The situation should exhibit a certain inertia because of the need for capital investment to change the configuration of rental types and the lack of a prospective market to make the investment rewarding.
If wealth becomes unequal, however, the better-off bidders can drive some of their competitors out of the market (because of legal restrictions on lower-cost housing) while obtaining increased area and amenities for themselves. It is to their advantage to do so, while it's to the landlords' advantage to accomodate wealthy bidders. (A strategy of accomodating lower bidders in housing with small areas per renter and poor conditions is not available, being prohibited by law). The result will be a less dense population per unit of land area or building volume.
Nevertheless, if wealth inequality were stable, a balance ought to be reached after all those those driven off the board disappeared permanently from the game. However, if the inequality of wealth were increasing, then one could expect continued falling density and shortages such as those described in San Francisco.
Rent control would probably not affect the situation one way or another (within the scheme of things presented here). This is because the exchange value of the housing unit does not disappear because of rent control but is diverted from the landlord and the banks, insurance companies, governments, lawyers, accountants, etc., which get money from the landlord, into other channels and places, e.g. the senior tenants. However, if rent control were abolished in a market where wealth inequalities were increasing, the rerouting of money towards the landlords and those who feed off them might be expected to make capital investment towards accomodating wealthy renters more interesting. So the abolition of rent control might result in a lower population density and an exacerbation of rental shortages.
In other words, the class-war nature of rental housing is not much mitigated by rent control except for some lower- class senior tenants (mostly at the expense of other lower- class people), but in a market undergoing transition to greater inequalities among buyers, it may slow down the development of shortages by inhibiting conversion toward lower-density housing.