My MIL is retired and renting. She only worked for about 20 years in her whole life, so doesn't get a full state pension. However, she gets pension credit (which is a couple of pounds a week less than full state pension) but it entitles her to full housing benefit (she lives in a council house) and full council tax paid for her.
Someone with a private pension wouldn't get that automatically, it would be means tested, and if they had a private pension would conceivably be little better off.
The issue of pensioners who pay rent has been a matter of concern to social welfare people for a while. Pensioners with private + state pensions are likely to find that they're little better off that pensioners on state pension alone, as a big chunk of their private pension will go on rent and council tax, especially if they are renting privately as rents are so much higher. And the benefit bill go up significantly.
I'm still working part-time, but am over pension age. I get a private pension of a bit over £300 a month and full state pension. I'm lucky enough to have paid off my mortgage, and when I stop work, the only help I will get is around £90 a month reduction in my council tax, and I will have all the costs of maintaining my home.
I'll be very jealous of people like MIL, who only have to pick up the phone when something in the house needs fixing!
For those nearing retirement and renting privately, it's worth considering registering with the council for social housing. It's much cheaper, and it's easier to get than most social housing as properties for older people become available a lot more often. One of my colleagues got a great HA flat after only about 9 months, and she was still working.