You are arguing as if buildling more make things worse, when in fact, this is not the case. The opposite of buildling more is building less. Building less will increase existing price a lot more on an existing neighborhood than building a new buildling will ever do.

When there's not enough housing for people to live in, people start competing on rent and bidding wars on purchases, this drives price very fast.

That's why you have to build more. To avoid the biggest issue which is a housing shortage.

Is that the ONLY solution? Of course not! Public housing is great. But the same things that prevent YIMBY policies is the same thing that prevents public housing to be built. When you advocate for more zoning to prevent new condos in your neighborhood, it's not just new condos at market price that will be prevented, it's a whole range of housing.