The USA has an elected head of state - and look at some of the people they got (Nixon, Carter, Reagan, Bush junior etc)
And often these elected heads of state are bankrolled by billionaires who have their own agenda.
Once the president is in power (be it Reagan, Bush or whoever) they start having to make good to the people who bankrolled them, passing business their way (like in Iraq for example).
While I think our royal family need bringing down a peg or two, and a few taken off the payroll, I dont believe an elected head of state is always the right answer.