Regardless of whether or not it as successful, the utility championing American values abroad to further the national interest goes back to the first great debate in American International Relations. Indeed, early US Foreign Policy was characterized a heated debate between Alexander Hamilton and Thomas Jefferson: Hamilton, Washington's Secretary of State and right hand man, was a Machiavellian pragmatist who believed the early United States should be assertive with weaker polities (and absorb them if possible) while appeasing stronger ones. Jefferson, on the other hand, believed in the moral superiority of the United States (the country being a Republic and all) and that the fate of the United States was tied not only to the fate of freedom in the world, but the world should be nudged towards universal republicanism through, "Peaceable Coercion."

In short, the general current of thought (also exemplified by a number of prominent thinkers at the time) stipulated that the more Republican governments there were in the world, the safer the United States and other republics would be. The debate, then, was centered around the extent to which the United States should actively foster global Republicanism.

Jefferson was a non-interventionist, who believed that if the United States went abroad looking for "Monsters to Destroy" it would become like the Roman Republic, with an increasingly influential army eroding democratic institutions. True to Jefferson's concerns (in addition to other limiting factors) the United States would have to wait for president Woodrow Wilson before the imposition of American democratic values abroad became a policy goal. Wilson, who okayed the US' decisive intervention in the First World War, arrived at the post-war Versailles Conference armed with "Fourteen Points" affirming the importance of democracy and self-determination on which he intended to formulate his demands for the peace treaty. His goal was to create an international order which would avoid another war in Europe which would drag the United States into the mix. Although Wilson was out-wiled by his more vengeful european counterparts (the French Prime Minister Georges Clemenceau stands out as a particularly tough cookie) his interpretation of international affairs has by and large been the strongest influence in modern American foreign policy (with peaks and troughs, of course).

After the Second World War, the United States created a number of international institutions (the IMF, the UN, NATO; just to name a few) making itself the "Indispensable Nation" and transmitting its values to the whole world. The United States also accelerated post-war reconstruction in western Europe through the Marshall Plan, making sure European states (politically and economically) embraced American values, and actively worked to contain the spread of Soviet totalitarianism. The United States was also one of the champions of the European integration process; seeped in the belief of the superiority of its own federal system of government. So you see, the end goal was always to be American security, however there was an entrenched belief that peace and security went hand-in-hand with the adoption of American values.

This all changed after the Vietnam war. The disastrous result of the conflict in Vietnam was, in great part, a failure in state-building. The "Vietnam Effect" made voting public as well as US government much less enthusiastic about coupling the spread of American values to a positive effect on national security. This was the reasoning behind Nixon's policy of "Détente" in foreign policy, which opened the way for a somewhat less engaged and more cautious US foreign policy. In all, the logical conclusion is that Tillerson, in the quote cited above, is referring to more recent events in US foreign policy that break the 20-year-rule here at r/AskHistorians.

A great overview of the historic roots of US foreign policy can be found in John L. Harper's American Machiavelli: Alexander Hamilton and the Origins of U.S. Foreign Policy, published in 2004. Another work by the same author on the United States' role in postwar Europe is American Visions of Europe (1994) while an interesting overview of the "Vietnam Effect" is treated by Frederick Logevall in his Embers of War: The Fall of an Empire and the Making of America’s Vietnam (2012).