I stated myself that she considers herself "servant" of the people, Britain!
But I protest against the opinion that she is a servant of any government.
And she is, despite being a powerless symbolic figurehead when it comes to daily politics, not a willingless puppet!
I strongly disagree when you say, that she has to do whatever the PM tolds her to.
Luckily there were not so many decisions by the government that she was so opposed against that she felt not to do what they asked her to with the exception of Mrs. Thatcher. Some papers reported back in the 1980s about the Queen being unhappy with some of the Thatcherist decisions; The PM got some bad press which didn´t contribute her image in a way she would hoped.
Another example: Mrs. Thatcher didn´t want the Queen to take part in the 1979 heads of commonwealth meeting in Lusaka. Downing Street communicated this very clear towards Buckingham Palace. The Queen said, Of course I will go, and that´s what she did! Mrs. Thatcher had no saying in that decision, just agree and smile.
Yes, the Queen is a constitutional monarch without power of daily politics, but she is very influential (like the monarchy itself) in Britain and voices her disagreement about issues (behind closed doors) she´s opposed to.
If the Queen had said clearly "No, I won´t open on june 3rd!", Cameron would have been, more or less, forced, to find another date.
If he´d say somethin´ like "The Queen won´t open herself this time because she´s too busy, strain too great etc., someone from parliament will take over", this would backclash against him, not the Queen! The damage of his chaotic planning of this state opening is already been done. But there´s only pity for the Queen and anger about the PM.