You're focusing on yet another externality of vacant units, that they increase the price of housing. I agree with you that this is minor compared to the way that restrictive zoning drives up the price of housing, but what about all the other negative externalities I mentioned in my comment?

Zoning for higher density doesn't necessarily solve this problem.

Creating more supply results in more units that can potentially be purchased as an investment and left empty.

You're arguing that vastly increasing supply will not lower house prices?

Not at all. Adding more supply is definitely part of the solution.

The problem is that adding supply takes time, there are physical limits to the rate at which supply can be added, and the level of demand is unknown.

This means it's hard to know when a supply only solution will result in decreased prices and it can take a long time. (Vancouver is not SF, it adds plenty of supply and yet there is still housing shortages and high prices)

If you want to fix the problem quickly, policies that increase supply and limit speculative demand will achieve results faster.

The problem isn't going to be fixed any quicker than the time it takes for supply to be increased.

Housing is a good investment because it's illegal to make more of it. This makes the price of existing stock go up.

If you legalized the production of housing that would no longer be true.

No, the fact that the whole lower mainland has 30% the population of NYC is why Vancouver proper has 1/3 the density of Brooklyn.

Increasing density in response to rising populations is great. Increasing density in response to rising investor demand is what we've done already, and it's why we have incredibly lonely, dead neighbourhoods like Coal Harbour.