In those areas that are primarily zoned single family, the problem is that you can't find something to buy. Not that you can't find somewhere to rent. Now if your talking about affordable homes to buy or to rent, again that's not because people aren't building new homes or new buildings. It's that there's no incentive to build affordable units in high desire areas. The whole point of building in a high desire area is that you can elevate the prices and your margins. If you're going to build an apartment building in a high desire area, you're going to build luxury units.
Why? Because there's no shortage of people who can afford them. You don't have to build low income to avoid vacancies.
"Allowing diverse kinds of dense, multifamily housing units to be built in the suburbs..." misses the point. The problem is that many people don't want dense, multi-family housing in the suburbs. They move to the suburbs because they want single family homes. Building more of what people don't want doesn't fix your housing problem, it just wastes money.
It is a profit margin conversation. Where do people want to live? Near their work or near good schools for their kids. Do people prefer single family homes or do they want to live in apartment buildings? Young professionals like apartment buildings more than older people with families. But young professionals are not as well positioned to buy a home than older people with families. That means that your buying market is primarily comprised of established professionals who are moving into the family building stage of their lives. Developers understand this -- it makes more sense to build a single family home with great margins and sell it unless they want to also take on the long term headache of managing a property with constant tenant turnover. And the flip side is to build luxury apartments for young professionals who are still going out and living that fast life but have the income to afford a better quality space.
I do a decent amount of landlord/tenant work so I've found myself looking into the rental markets more than usual. There are plenty of low income spaces to rent in most cities. The neighborhoods suck, the amenities are non-existent. But those landlords will take almost anyone with a job. The rental housing is there. But the people in the buying market aren't.
When someone says "We should just build more apartment buildings," I often wonder who they think prefers to buy those premises over single family homes? And this matters because the housing crisis is almost exclusively discussed in the context of home buyers, not renters.