Yes, our system is FAR FAR superior to your system of government. To be honest you are just very lucky your country is very liberal with their personal freedoms. Any version of "monarchy" is a horrible horrible system. Also you have a state run religion which is another horrible thing and why our 1st Amendment was written. I don't want to start that argument here but thats what this fictional "seperation of church and state" is all about.
Based on what do you draw your conclusion that the US has a vastly superior system of government?
The system of constitutional monarchy has stood since 1689 and has not been plagued by any Civil Wars. The United States has only stood under its present system since 1787 and has suffered one civil war.
We know how to conduct elections that don't have stupid things like pregnant chads involved.
The British judicial system is incredibly politically independent; We don't get any of this political bull where people get appointed to the Courts on basis of their ideology, but rather on their intellectual ability to do the job.
Our Head of State is widely respected by most citizens; Even republicans speak well of her. Your Head of State, regardless of whom it is, often gets derided by half the country. A Head of State who stands above the political fray is inherently beneficial as it creates a symbol around which national unity can be created.
Whilst the US has an advantage in terms of protection of freedoms, I would not say I consider myself any less free than the average American. Britain has long standing Constitutional conventions that could not be ignored lightly. Also remember, that most of your political freedoms were simply carryovers from the UK and that we did indeed invent the modern conception of representative democracy.
As for the State Church thing: Yes, we have a state church, but the Church knows full well that its role in society is not impinge into the political sphere heavily. Whilst there is a State church, little law is based on the purely religious values declared by Christianity, and in fact much law stands opposed to the text of the Bible; For example, sodomy is legal. The existence of a State religion is simply nothing more than a continuance of a tradiditon, that whilst not something I would desire if starting from a blank sheet of paper, is not something that does the country any harm in particular. You are incorrect however to assert that the religion is "state run" - It is not: The Church of England is governed totally disjoint from the UK government.