This 2008 lecture by a brilliant Rhodes Scholar and Harvard law professor discusses why large majorities of the Muslim world say in opinion polls they want shariah law.
He describes it as a legacy of shariah to restrict power on the executive (the caliph) and establish rule of law in the Muslim world. — Much like English nobles founded Parliament to check power on the king. — So when Muslims say they want sharia, they want a check on their autocratic dictators, protection of property and individual liberty from the State.
He also worked for the US CPA during the Iraq Invasion in 2003, and speaks candidly about his experience with Muslim democracy with the US adventure in Iraq.