Something mirroring the contemporary British system was actually proposed at the Constitutional Convention and shot down.

Let's set the scene for what the country looked like in 1787. Before the War of Independence, the state governments had been largely autonomous, and there was very little tying the new country together aside from their united antipathy to the British. State loyalty was MUCH stronger then than it is now. As late as 1860, half the country decided that treason en masse was preferable to disloyalty to one's state.

Into this atmosphere, you have Alexander Hamilton's plan, which would essentially replicate the British government on this side of the Atlantic: a popularly elected House of Representatives, a Senate chosen for life by the electors, mirroring the House of Lords, and a Governor appointed for life by the electors as a sort of an elected monarch. State sovereignty would be eliminated. (Hamilton's biographer, Chernow, adjudged this plan to be "brilliant, courageous, and in retrospect, completely daft.")

This was a non-starter, as state governments were unwilling to let the national government eliminate their sovereignty. Instead, the current system, which splits authority between the state and federal governments, ended up being the compromise that everyone could agree to.

Chernow, Hamilton; Madison, Notes on the Debates in the Federal Convention; Hamilton, Madison & Jay, The Federalist.

Can you give some examples? The only real difference I see is that the head of government is directly elected in the US vs. elected from the MPs, or am I missing something else?

/u/CiderDrinker gave a good short answer yesterday in the "shadow cabinet" thread. Basically, the modern parliamentary system hadn't yet been invented.

Here's a similar question posed three years ago.