Balderdash - unless you count “being able to work” as a luxury.

Public transport in London is painfully expensive, and if you are earning minimum wage on a part-time contract, a private tenancy that would be affordable to such a person would be well outside of the city, and their commuting costs would be a substantial portion of their pay - and you would need to subsidise them to a greater degree than you would by simply having them live near their place of work.

So, again, I assume you count having a job and a home as a “mountain of privilege”, and not as a fairly universal underpinning of society.