>bleak economic prospects of structural inequality
The housing crisis has nothing to do with economic inequality. If there are 3 units and 5 people who want them, 2 are going to be dissatisfied, even if they all make the exact same income.
Lower inequality just changes the allocation strategy away from "who has the most to spend" to things like government lottery, who was here first, who belongs to the most politically favored class, and who is willing to spend the largest share of their equal incomes on rent (making the most sacrifices in other areas of life). Arguably, we already live in this world, due to SF's progressive policies around housing allocation which seek to dampen the influence of "how much do you have to spend on rent" on what (if any) housing you'll get access to.
The housing crisis would remain exactly as severe as it is now if you erased 100% of inequality between people.
We have it because there is economic inequality between geographic areas, leading people to migrate to areas where this no room for them, and no will to build vertically to make room.
But that's only in a world with flat population. If we were reproducing above replacement rate, a housing crisis would be inevitable without construction, even if every person and every area had exactly the same socioeconomic status.