2016 Postmortem
In reply to the discussion: Hillary should have Bernie as a running mate. [View all]Our Speaker of the House can wind up being a coalition leader, just as a Prime Minister often is, if some rules changes relax the two-party hold on Congress. The President's cabinet could easily be made up of Members of Congress, for it is the Senate that consents to their appointment and they can simply deny non-Members (as they often did in the 1800s). That in turn would make the Cabinet much more like Ministerial positions. Some of those departments, like the Department of Justice, are largely autonomous and outside of the President's control already.
Much of the President's power is delegated to him by Congress, though changing that now would require actual changes to the law rather than simple changes of rule and precedent.
Our system grew organically from the British parliamentary system, which is also just a few rules changes away from moving away from two-party rule.
Edit: I should add that Congress already is made up of coalitions. The Republican Party is being devoured from within by the Tea Party, while the Democrats also include independents such as Bernie Sanders and semi-independent parties such as Minnesota's Democratic Farmer-Labor Party.