They're not privileges, they're performing a service needed by the country.
The NHS paid out £2.3 billion in damages due to the failure of obstetrics. Would you put right what's going wrong for a fraction of the cost or privatise it?
I'd improve the BBC by cutting its links to political parties. If you've worked in a government position you're not impartial, and the chances are you're no longer in that position because you're not very good. You shouldn't then be placed in a senior position in an institution such as the BBC to either provide a service to suit your party or be intentionally positioned their due to your incompetence.
Maybe you're looking in the wrong place for the villain.?