If there isn't enough housing there's no way to make it affordable outside of the government buying the buildings and instituting rent control, which will never work given how much hate imminent domain gets and how inefficient the government is at managing literally anything.

Building more needs to be a part of the solution, but what if you built only market-rate housing and applied subsidies to existing housing?

Build old / used apartments of course.

1. If there aren’t enough cars to go around, someone’s going to need to make new ones.

2. Real estate doesn’t depreciate like cars do.

to help with this analogy - consider that for every thousand of who needs a car there is only one hundred cars existing at any given minute. Some new, some used. Among those 1000 some have more money, some less. Solution here will not be just start looking for used car, there is a different problem.