You are in effect stating that AF is guilty because she was thought guilty and thats enough in 'realpolitik' terms, but to couch political reality and good governance in such Machiavellian terms is an opinion not a fact and cannot be seen as a basis upon which to convict someone.
I started this thread to analyse what AF actually did or did not do. Not to analyse what people think she should have done, thats a valid but nevertheless separate topic/debate.
Your example of Cameron and the British private parliamentary secretary is a perfect demonstration of coercive and amoral judgement being used by a political leader in relation to public opinion, yet you are stating this is exactly what AF should have done with Rasputin in your opinion.
It is precisely because the accusations being made against Rasputin and AF were unfounded that they chose to defend him against what amounted to little more than hysterical bigotry agitated by their known enemies both internal and external. To concede to such demands and from such sections would have set a very dangerous precedent. Bear in mind Rasputin held no official public position, he was merely a personal friend, therefore his relationship with the Imperial couple was not, strictly speaking, a public matter.
People were being deliberately led to believe things such as the Polish Capitulations and millions of troops sent to their deaths unarmed and barefoot was not down to GD Nicholas N's arrogance and ineptitude but because the 'lovers' Rasputin and AF used a secret radio transmitter on the roof of the Alexander Palace to communicate state and military secrets to the Kaiser! I fail to see why you see such perceptions as a basis upon which to advocate that N&A were guilty of poor judgement for not giving this type of accusation credence.
A contemporary example would be advocating Obama denounce his Kenyan Muslim heritage and remove his middle name Hussein, because hysterical and bigoted elements of US society see it as proof that his administration is in league with Al Qaeda.
Rasputin, rather than demonstrating any supernatural prescience, accurately gauged the biggest threat to the Tsar when he predicted that if he (Rasputin) was murdered by members of Nicholas II's family then they would all be dead in two years; ie he understood (more so it has to be said than the likes of Lenin at this point) just how vulnerable the regime was to attack from within and that Nicholas II's 'achilles heel' was his inability to discipline his own family from agitating against him.
Public opinion of this, after Rasputin's murder, unlike the campaign against AF was based on reality and so did far more harm than what came before as it showed the regime to be riven by internal conflict that the Tsar was unable to control, and he was therefore defenseless from attack.