Housing-first is a great way to deal with homelessness, because most of the problems homeless people have in rebuilding their lives are compounded by being on the street. I’m not saying we shouldn’t house homeless people.
I’m saying that comparing the vacancy rate to the homeless population is ridiculous, and isn’t evidence that there’s no housing shortage.
Partially, that’s because vacant houses aren’t all habitable, or able to be sold/rented immediately. But also, it’s because having some number of empty units on the market ready to be moved into is a good thing. You don’t want to have to find someone who wants to move out the day you want to move in. That creates a sellers market, causing high prices.