I don't work in the public sector, I'm an end user, so I don't really have any insight into the inner workings of the NHS or the police.
However, whilst it's not universally agreed, it's generally accepted I believe that the NHS satisfaction rate has gone through the roof. As a user, I can certainly say my experience last year was incredible and a far cry from when I had to go through a similar process in the early 90's. I have a breast care 'nurse' who doesn't wear a uniform, she doesn't do the rounds of a ward, she has an office. She is the single most important person I have access to whilst going through this.
Same with the police. Statistics seem to suggest that crimes rates have fallen significantly. Again, it's no doubt open to dispute, but certainly we seem to feel safer.
I worked in the private sector, in a large multinational organisation, tens of thousands of employees, the majority of who didn't actively deal with the product we sold, but were the back up, the structure that let those on the frontline get on with their jobs.
I'm not suggesting that there isn't a case for slimming down or that things are perfect, but I'm not sure it's as simple as saying we need nurses who nurse.