I know what you're saying but I don't think it's really true, at least not in all sectors. It's easy to demonise the contractors but are you truly sure they're much more expensive than a full time employee, once you take NHS benefits, employer overheads and short v long term expenditure in to account? You should make sure what you're saying is true before demonising private contractors, although I understand it's a tempting and easy thing to do, especially if you are on the public sector side of the fence and have anecdotal evidence of it.
I work for a strategic partner, providing services for another strategic partner, who in turn provide services to the public sector. I do this alongside a team of public sector staff, from which I came from, and I can tell you that - using your fat/skeleton analogy - the place is positively obese across the board. I've seen it from and continue to see it from all sides, and it's definitely absolutely not "public sector good, can't pay people enough, private bad".