RE: She has no direct control of the process.
In other words, her role is simply to rubber stamp the process, even if she personally disapproves of those nominated?
If the answer is yes, why can’t they just replace her with a robot?
Yes, constitutionally she has to act on the advice of her ministers in all public acts.
But, She does serve two vital functions. As an impartial, non-political Head of State she "separates patriotism from politics". The Queen and the Royal Family, by being a living symbol of the constitution and the focus of national celebrations and mourning, prevents great and small ceremonial events from being politicized. She acts by her existence to give politicians some humility. The Prime Minster may be able to lauch nuclear weapons, set national policy and even modify the Constitution, but he just a politician, and must be respectful to the Queen.
Also, by virtue of her reserve powers she is the last resort in a constitutional crisis. In a true crisis, with the legitimacy of the goverment in doubt, she can dissolve Paliament and force an election, sending the issue to the people to decide.
A robot wouldnt look as good in that hat :)