But he doesn't have unlimited or extra-Constitutional authority either. If the Framers wanted the President to have dictatorial powers during war they would have explictly put that in the Constitution. They didn't do that however. The Framers more than anyone understood the balance needed between the legitimate needs of government and individual liberty.
They wrote the Fifth Amendment requirement of indictment for example, even knowing that in some cases, conspirators could be tipped off by the simple fact that an investigation was occuring. They rightly believed this to be a better outcome that simply hauling people into court on flimsy evidence. Ditto for the Fourth Amendment - conspirators may be tipped off by a seach and destroy valuable evidence before the government gets to it - but the Framers thought this was a better outcome than giving the government the power to randomly enter your home.