"The whole will of democracy is imbued into the president" is some insane, general will, Robespierre nonsense. The president's main job, under the Constitution, is to execute the commands of Congress. In almost all areas - including war and trade - Congress is supposed to determine what policy will be, and the president is supposed to carry it out.

Unitary executive theory is also wrong, because Article I explicitly gives Congress the power to pass any laws it deems necessary and proper to carry out its commands - including laws that, for instance, establish independent agencies. But even if you believe in unitary executive theory, that does not change the fact that the executive branch's chief function is to execute Congress' commands.