Why cant the public sack professionals in public institutions?

Why is it impossible for the public to sack incompetent professionals within public institutions?
Hardly a day that goes past without me wondering how on earth any one of a number of individuals working for any of the large public institutions could have possibly risen to their position of responsibility based on ability.
On the grounds that it would be very hard if not impossible for the public to monitor the rise of these professional dolts within these institutions it would seem necessary to have a mechanism in place where the public can easily sack these individuals on our terms and not theirs when problems arise.
This would be reasonably easy to achieve using a website.
This is not a one dimensional attack on professionals, indeed the net effect would be to improve the working experience and promotion opportunities of all competent professionals as they would not have to deal with the substandard ones or clear up the mess the dolts leave behind.
As it stands these professionals have created their own protection mechanisms within their various institutions and professional bodies in a completely non-transparent way.
We need a system for the public to identify, report and discipline various professionals, which would include the introduction of independent members to any profession's disciplinary panel.
Until then there will remain a number of overpaid under IQ'ed professionals who disproportionately bring institutions down to a dysfunctional level.
To which professions and institutions would this apply? Any which have chosen to protect themselves with any of, a lack of transparency, an ineffective disciplinary process or complaints procedure.
The NHS would be the obvious choice but you could apply the above to virtually all public institutions.