The reality is though that intergenerational wealth transfer are the primary way people acquire the necessary equity to buy homes (they get help from Mom and Dad). So I think you need both... both higher incomes, and mechanisms that create intermediary stages being renting and owning.
And on being a stretch, don't disagree. I feel the incentive is to be provocative beyond reasonableness, so apologize for that! Public schools in America are suffering death from a thousand cuts. I will tell you though that when urban school turnover (largely as a result of rising housing costs) is now up to 25%/year... it becomes your primary issue pretty fast.