The monarch's job is to continue arbitrary and unlimited power, and hand most of it over to government. In 2003, Blair used this to declare war on Iraq without prior parliamentary approval, in 1992, John Major used this power to cover up Britain's arms trade with Iraq, and in 1984, Thatcher used it to prevent civil servants from joining/forming trade unions.
The Queen has formal powers too, such as the power to dissolve/dismiss government, withhold royal assent on certain bills, and to appoint Prime Ministers. Even though they haven't been used in Britain for a while, they were used against democracy (with great backlash) in Australia to dismiss the 1975 government, and in 2008 to prorogue the Canadian government (and undemocratically prevent a vote of no confidence) for several weeks.
It is true that Elizabeth isn't exactly controversial, but monarchism ends up choosing people by chance, and there's a danger people like Charles could abuse it. Additionally, her being above the fray of politics is pointless, when she ends up being the government's puppet. This is where the equality of opportunity argument comes in (and yes you can think it baseless in comparison to the white millionaires club), it's not good enough for me to just say that the current head of state gives birth to the next one; I think such a position should be earnt.
