I agree that so-called market rate housing is not affordable. Let's consider why -
1) typically these market rate housing in new developments require the market rate owner to subsidize someone else's rent under 80/20 rules. So of course, if you pay your rent and someone elses, that starts getting expensive
2) many places (East Harlem is a great example), constrain the available supply of land for development by refusing to tear down projects and build high density. The truth is most projects are low density antiquated buildings compared to the housing the market place would build if we zoned those neighborhoods R10.
3) rent control is a major disincentive to new development and artificially constrains supply. People claimed in Cambridge, MA (a built up, expensive town) that eliminating rent control would push out the poor - but that didn't really happen. Instead, investment and supply rose and while the average income rose, rents have stayed pretty stable.
4) Chicago/Toronto disproves the theory that density requires high prices. If you allow supply to grow to reach demand, prices stay within reason.
Will Manhattan always be more expensive than the outer boroughs - yeah probably somewhat. But the differential is exascerbated by bad public policy that discourages investment on the theory that affordable housing increases affordability - its not efficient, creates scarcity, and those undermine any theoretical advantages from high subsidies.