We're not ruled by any queen, Maestro. Like I said, and you know this anyway, it's strictly for ceremonial purposes. Same goes for the Governor General and the Lieutenant Governors. Don't put too much belief in that "de facto head of state" crap you see on the government websites and stuff. The role of the GG is no more real than that of the queen. No, unfortunately though we just pay taxes to support a completely useless sponge (Canada's Governor General that is, not QEII). Also, whether you choose to call a leader a prime minister or a president makes no difference whatsoever. They perform the exact same function either way. That in and of itself doesn't tie us to Great Britain, it's merely a title. Lot's of non-Commonwealth countries have prime ministers these days.
Incidently, not all present-day Commonwealth countries call Elizabeth their queen. India, Pakistan, and South Africa would be prime examples.