I think people are talking past each other here because "Sunni vs. Shia" is a distinction that has great meaning to those on the inside of that religious or regional context, but means absolutely nothing to most on the outside - especially but not only those in the US.

Imagine that there was a wave of Buddhist terrorism. Would people in mostly-Christian or mostly-Muslim countries notice or care about mahayana vs. hinayana (vs. other)? Big nope. If they made any distinctions at all, those would be on the basis of country/region or skin color or just about anything besides doctrinal differences. More likely they'd just tar all Buddhists with the same brush. Do people in Iran pay attention to factional differences in the US, or are we all "Americans"?

So it is with US vs. Iran. Sunni vs. Shia might well be important, might be highly visible to you, but to at least one half of the people making geopolitical decisions it's just not part of the equation. Maybe that's a good reason not to have the ecclesiastical and secular authorities so intertwined, as it makes such confusion almost inevitable (cf. Israel vs. Judaism).