If a few thousand airbnb units cost renters that much money, imagine how much renters would save if more and taller buildings were allowed to be built!
11+ super-talls (300m+) and 40+ talls (150m+) are under construction at the moment, and more than that has been completed in just the last few years. Massive amounts of rental units and co-ops are coming online in UWS (Riverside Center), Hudson Yards in Mid-Town etc [1]. And prices have come noticeably down as a result, you can also get more concessions than ever. I just recently received 3.5 months free on a 14 month contract in a brand new building.
[1] https://ny.curbed.com/maps/nyc-new-development-hudson-yards-...
https://ny.curbed.com/2018/1/24/16925172/nyc-rent-2017-decre...
But even at a record breaking construction pace, we're building <20K units annually (that's 50 400-unit buildings) while population is increasing by >100K annually. And that doesn't even include the people who would move here if it were less unaffordable.
If you look at the rezoned mixed use districts like the garment & flower district and the lower east side, they've been flooded with hotel development (in some cases clusters of hotels on the same block) because it's more profitable.
In addition to actual height restrictions, neighborhood associations also fight tooth and nail to prevent new development and to limit the height of new development when they can't.
[1] https://www1.nyc.gov/site/planning/zoning/districts-tools/re...
There are a few historic districts (read: well-off NIMBYs) such as Park Slope where height is strictly capped.
In Manhattan the caps are generally designed to ensure that at least some light and air still reaches the street (before they were imposed, buildings were build to their full height to the edges of lots, and contributed to the city being very dark.
Much of Manhattan is actually somewhat low-rise (3-5 stories). In those areas, developers can buy "air rights" from neighboring low-rise buildings if they want to build taller on a nearby lot.