Expensive housing is also correlated with better provisioning of services which draws people here.

I am not redefining anything, I am describing to you what people are colloquially talking about and why these conversations oftentimes are just people talking past each other because they have different issues in mind. When people in SF say "homelessness is out of control", they are typically talking about the population I am mentioning - one that is unlikely to be helped by changes to housing costs at the margin.