We do not need a democratically-elected Head of State provided we have a democratically-elected government to pass the laws which govern us.
The Queen does not govern us, and we "bend the knee", bow or courtsey to her only literally and if we meet her (that is if we have more manners than Cherie Blair).
The Queen's job is to stand head and shoulders above the sordid world of politics, to represent to the outside world everyone in the country regardless of race, creed, colour, sex, political views, tastes or social status. No mere politician could possibly do that.
(PS: As many on this site know too well, I am not religious!)