I don't see a lot of people suggesting we can legislate our way out of housing prices without building more housing stock.
What I do see is a lot of what amounts to saying that "those people" should just go somewhere else. As long as that somewhere else isn't one where I have to notice their homelessness.
But yes. We have to build upwards. In cities where building vertically benefits the well-off (as in NY), and where it's profitable to do so, it happens. But note that it only generates high-end and expensive housing stock (excluding rules that require including "affordable" housing, which based on the examples I've seen seems to mean "affordable to a family making $120k/yr instead of the $500k you need to afford the market rate units in this development".)
Will adding more housing stock at the top decrease prices at the bottom via market effects? Yes and no. If the availability does trickle down, it takes a very long time.
Here in downtown Brooklyn there's a temporary glut of high-end housing added in the last 5 years. But compared to the number of people added to what was already a tight rental (and sales) market in the city 10-15 years ago, the number of units added is laughable. We're talking almost a million more people living in the city but something like 20-30,000 units (enough to house 50-100k people) having been added.
So what happens? This oversupply will likely last about 5 years. The only real effect will be 1-2 year 10-15% concessions on rent in the new rental buildings until they can fill them up. People looking for high-end rentals will be able to take 24 hours instead of 3 minutes to commit to a specific apartment after viewing it, and might be able to get it if they wait an hour or two to apply instead of contacting the management company within 5 minutes of the listing going live. In a couple years, the high end rental market will go back to its normal cut-throat self.
Maybe the gentrification of Crown Heights will be delayed by a couple years, but I doubt it. This trickle of new housing at $2500-4000 per person per month isn't nearly enough to keep up with the constantly expanding demand for units under $1200 per person per month.
The for-sale units in the crazy high-end buildings will sit vacant until enough new Chinese billionaires are minted who want to park their money outside the country, and then they'll continue to sit vacant indefinitely anyway because they're investments rather than homes. The ones in only normally high end buildings will also sit vacant for a year or so and then generate a tiny bit less profit than the developers were hoping for. But sales don't really factor into the affordable housing market here anyway. If you're having trouble affording rent, buying is unimaginable.