Many people live in areas where the benefit cap makes a huge difference, and they have to use money supposed to be for living expenses towards the rent
Yes but the rent is based on the average for that style of property in that area. The average. For their specific area.
OP, you are conflating the benefit cap and the local housing allowance (LHA).
The benefit cap is the same all over the country except for London, where it's higher. It's the same in Bassetlaw as it is in Brighton, but rents in Brighton are 2.5 times those in Bassetlaw. So if two benefit-capped families have identical circumstances other than geography, and therefore rent, the family in Brighton will be far worse off because they will have to pay a lot more towards their rent.
And the local housing allowance is not based on average rent for size of property, but on an average of the bottom 30% of private sector rents for that size of property. In years when CPI inflation is less than the rise in rents, the increase in LHA is restricted to CPI, so it isn't even an average (thank you for that one, Osborne and Cameron) So the shittiest, most run down of properties in the roughest areas are the only ones that are affordable for poor families.
Even so, in much of the SE and other high-rent areas, it is near impossible to find a private rental at or below the LHA rate. Rightmove currently has one 3-bed property that's not above the LHA for the Brighton Broad Rental Market Area, which extends from Shoreham to Newhaven and a good 10 miles or so inland from the coast. People on low incomes are forced to live a long way from the main centre of employment, so if they are in work, they have to pay significant commuting costs and cannot take jobs that involve working unsocial hours because there would be no public transport available for some shifts.