Why do undergraduate admissions in the U.S take into account nonacademic criteria?
As with any complicated social phenomenon, there is no simple and conclusive answer.  However, here are two important factors:


The U.S. didn't have world-class universities until the 20th century.  Even ones that are world-famous today (such as Harvard or Yale) did not particularly impress 19th century Europeans.  These universities had started out as vocational schools for pastors and gradually turned into finishing schools for the elite, but they weren't scholarly powerhouses.  In the early 20th century, there started to be more academic competition for admission.  This was very upsetting to the traditional students (largely wealthy young men from prep schools), who didn't want to be around too many Jews or other minorities or to have to compete with nerds for grades.  One tactic universities used in response was quotas for Jews, but the Nazis made that look bad.  Leading U.S. universities then moved on in the 1930's to develop other methods to ensure that they could pick whichever students they wanted.  For example, geographic diversity (you should take students from Kansas to avoid having too many New Yorkers), preferences for children of alumni, athletic recruitment (you really need a strong lacrosse team and fencing team), well roundedness (students should study hard but not be too nerdy), extracurricular activities, etc.  Jerome Karabel has documented this history in his book The Chosen.
What is Harvard's purpose in educating students?  People often imagine the goal is to educate smart people, and that corruption is the only explanation for why Harvard would deliberately admit a wealthy applicant over someone more talented but poor.  However, this is thinking about it completely wrong.  Harvard's primary goal isn't to educate smart people, but rather to have an impact on the world by educating influential people.  They want to educate the people who are going to grow up to become leaders of all sorts (social, political, commercial, academic, etc.), and while they are happy to help shape who becomes a leader, they know perfectly well that wealth and privilege play a major role.  When Harvard admits someone whose wealth exceeds his intellectual talents, it's because they want to help set the national agenda by providing this person's education.  From this perspective, all the strange admissions criteria are an excuse for Harvard to select whoever they feel has the most potential to change the world, taking into account all aspects of their talents and background.


So what about other universities?  The first axiom of higher education in the U.S. is that everybody imitates the most prestigious universities.  If Harvard, Yale, and Princeton are doing something, then everyone else will follow.  Not necessarily in exactly the same way: different universities may employ very different criteria for admission, thanks to different goals regarding who they want to educate.  But they almost all use the same basic framework for what information is relevant to their decision.