Seems like you would probably get on just fine with a fabricated character played by an actress, who is charismatic enough. Say, "Big Sister"
I'm not having a pop at Americans, but it may make sense to view the queen as a symbol in the same way that the flag is for Americans. It's just that our symbol happens to be a person.
Also, the Queen ties us back to stuff like Henry V and Agincourt etc.
If I am feeling particularly generous I tend to agree with Samuel Johnson that "Patriotism is the last refuge of the scoundrel."
The rest of the time I am more aligned with Ambrose Bierce's version from The Devil's Dictionary.
“Patriotism, n. Combustible rubbish ready to the torch of any one ambitious to illuminate his name. In Dr. Johnson's famous dictionary patriotism is defined as the last resort of a scoundrel. With all due respect to an enlightened but inferior lexicographer I beg to submit it is the first.”