from BW
The majority of Shia populaces are not in states that have governments formed during the Cold War by western governments that shaped the politics of the region to deny this critical Cold War battlefield to the Soviets. The majority of Sunni populaces are in states that are.
We have from COL Lynch's report this snip (p.47 - see also Table 3, p.49):

Only five countries (Iran, Iraq, Lebanon, Bahrain and Azerbaijan) have more self-identified Shi’a than Sunnis. Another ten states have Shi’a populations or diasporas numbering more than half a million and that hold
a politically important minority position. [81]

[81] In addition to the five Shi’a majority countries, Afghanistan, Saudi Arabia, Yemen, Syria, Turkey, India, Pakistan and Albania have Shi’a minority populations with significant numbers and political import. One should note that despite control by an Alawite, secular Shi’a leadership, Syria is a country with more Sunni than Shi’a.
Iran, Iraq and Lebanon were very much Cold War political battlegrounds, from the time of Kim Roosevelt on. Azerbaijan is independent because of the Cold War. I also fail to see the direct materiality of the Cold War to Lynch's thesis.

PS - COL Lynch, wherever you are. Please come on and explain your take to these other O-6s.