Why did the leaders of the young United States replace royal rule with a republic?

America's founding fathers intended the U. S. to be a Republic (elected officials vote on laws), rather than a Direct Democracy (everyone votes on laws).

More specifically, the founders intended the U. S. to be a "mixed-republic" comprised of a union of states (federalism), each with a republican government, ruled by elected officials and laws, bound by federal and state Constitutions, in which democratic and liberal principles were ensured. Indeed, they succeeded, and today the United States is a Constitutional Federal Republic; with a strong democratic tradition.

TIP: See also, "what is a Republic?", "Plato's Republic", and the U. S. party names explained.