The premise of this question is rather flawed. In democratic Latin America, Eastern Europe, Africa and Asia, Westminster-style systems are the distinct minority. Far more common in the aggregate are three other systems: strong Presidents (i.e., not only executive Presidents in the US sense but possessed of significant legislative power exercisable by decree), US-style systems, and French-style hybrids, with an executive President who shares authority with a Prime Minister and Cabinet somewhat or entirely responsible to the majority of the lower house of the legislature.

Where Westminster-style systems prevail, it's for one of three reasons. Mostly commonly, because the state was a former British colony, felt comfortable with that system, and never had a post-democratization crisis which required (or permitted) a strong Presidency to take shape. Secondly, because the state wanted to become a constitutional monarchy (such as the Commonwealth states which retained the Queen, or Japan or Thailand) and the Westminster system is the obvious solution to the state having, by definition, a head of state who is both non-elected and required to be above partisan politics. Third, pretty much only in Eastern Europe, because a state had a legacy of mistrust of authoritarian leaders and wanted to emulate the Westminster style systems prevailing among their western European (new) peers.