The current scarcity of affordable rental housing -- and when we talk about a shortage of any kind of real-estate, we're always qualified by 'affordable' -- is greater because of two trends.
The first was Hope 6 from the 1990's in which large public housing projects were torn down and replaced with much smaller and prettier public housing projects and by which the stock of public housing in the US was greatly reduced. Displaced tenants were expected to find housing in the private market using various vouchers, such as Section 8. This increased demand raised rents and removed surplus from the affordable housing market.
The second trend however, effected the rental housing market more broadly. In the housing boom of the early 2000's, a large percentage of the existing rental housing units in the US were converted to condominium ownership and the units were taken out of the primary rental market. Because of the way condominium insurance is written, many condominium deeds restrict rentals of units in order effort to limit association fees. This makes it less likely for former rental units to return to secondary rental markets after a property has gone condominium.
The issue is quite simply that rents don't cover development costs in much of the US for large demographic segments of the population. In part this is because the minimal cost of development is lower bounded, and not primarily by government controls but by land and construction costs established by the free market.
In the extreme for illustration, even if you get the land for free in Detroit, you cannot charge enough rent to cover construction if you build for the economic segment where most of the demand exists, and in the segments where the numbers work out, there is limited demand and every other developer is targeting that segment.
One last piece of rental housing development is also worth bearing in mind, the upfront costs tend to be higher and the payback longer than with owner occupied housing development. There's no deposits or presales to fund partial development or create an initial income stream. There's no revenue until the rent checks arrive so projects get built in a single phase. That means a bigger loan and an emphasis on faster construction times versus a slower more variable pace tailored to sales velocity.
Addendum: Another factor which may be coming into play is that rental housing has traditionally attracted institutional investors. To the degree that startups offer an attractive alternative the pool of potential investors for rental housing development might be reduced.