London does need a lot of workers, more than it can reasonably house. When I was still working (2 years ago, so not a lifetime!) - all the 'middle income' staff (the secretaries, reception staff, IT support, HR, researchers, those on between £30-45k) who had DCs lived outside London. The Partners were in London, the cleaners and catering staff were in London, the younger support staff without DCs were living in flatshares or in small flats with their DPs, but staying in London after having DCS seemed to be a choice for the rich or the poor, or accepting living in very cramped conditions.
These people, the huge army of people who commute in each day from across the SE to fill London's staffing needs, these people who've started again in new areas, faced the question "can I afford a 3rd baby", who've sat on trains an hour each day, these are the people who would have to pay more tax to stop benefits from being capped which will push poorer people out of the city. The Tories won because they won over people like this.
The cap is clever, setting it at the equivliant of £29/30k is hard to argue against when that's still above the average wage. The problem is, it's not cheap or quick to move somewhere else, and it takes organisation and effort to research new, cheaper areas. And if you aren't the sort of person who can get themself sorted to get a job in an area with so many job opportunities, you probably aren't the sort of person who will have the capacity to think about moving somewhere new and starting again.