Democratic movements have largely internal origins. The fallacy that has influenced American foreign policy since at least the time of President Woodrow Wilson is the idea that all peoples everywhere want democracy. The quest for individual freedom has been a strong one throughout history, but that quest is not exactly the same thing as desiring democracy. Moreover, the aspirations of people are, in part, a function of a society's structure (institutions, economics, culture, etc.) and history. The fundamental Sunni-Shia divide is part of the reason one has witnessed illiberal regimes in the Middle East. The peoples see things as a zero-sum game where one faction can only gain at the expense of the other (no "win-win" conceptions) and that strong rule is required to prevent societal fragmentation along sectarian lines. Western idealism assumes that authoritarian rule is largely an accident and that "regime change" can, therefore, quickly allow for democratic forces to take hold. The evidence has not been kind to that assumption.
There's also a tendency for the U.S. to view others as we view ourselves. Hence, the sectarian uprising that followed what had been protests for democracy was quickly lumped in as a democratic revolution in the tradition of the American revolution, democratic yearning in Eastern Europe during the Soviet era, etc. The reality is that a repressed majority was simply seeking to gain control over a brutal minority-led regime. Consistent with ethnic conflicts, brutality was in abundance and liberal ideals concerning humanitarian protections were discarded to the greatest extent possible.
Regional uprisings were quickly coined the "Arab Spring" in an analogy to the democratic Prague Spring. Not surprisingly, given the region's structural and historical context, the democratic illusions have proved largely unfounded.
The focus on national interests and strategic allies would preclude the use of force in such situations. Diplomacy and other non-military programs could "test" possible democratic aspirations, give support to genuine movements, and limit the risks should those movements prove less than democratic.
In terms of Saudi Arabia, among other non-democratic states, the U.S. has to deal with the world as it is. The U.S. can't dissociate from dealing with such governments when U.S. interests are at stake. To do so would simply be the other side of the coin of military interventions in the name of ideals. In this case, the U.S. would refuse to engage in relationships in the name of ideals. Both approaches are extreme polar opposites. Constructive relationships are often necessary in advancing the nation's interests and promoting stability. Of course, the U.S. can and should use its soft power to encourage improved human rights, etc., and influence the factors that might lead to a more favorable climate for democracy. The latter would require a lot of time and patience, as societal structures evolve slowly.