I can't speak to why other countries made the decision to use one data point above all others but can say the reason American colleges and universities don't generally boils down to the American belief in meritocracy and the diversity of the higher education system.
Each of the seven Colonial Colleges had their own admissions criteria (more on them here), although there were overlaps as they all focused on the same type of education and were shaped by the founders' experiences with and understanding of British universities; they offered a classical "educated man's" curriculum. That said, there were some distinctly American touches from the very beginning. Students (all male, all white save a handful of exceptions for Indigenous men) were accepted not only because they passed the entrance exam, which was more recitation than pen and paper, but because they could afford tuition, and were deemed sufficiently worthy of carrying the name of the College. This notion that a young man's character was as important as his academic abilities was linked to the emerging notion of America as a meritocracy. In other words, even if a young man had no family connections, if he was sufficiently ambitious and had the academic abilities, he could attend.
Kett's 2012 book, Merit: The history of a founding ideal from the American Revolution to the twenty-first century does a really nice job exploring how this idea of meritocracy played out in the colleges as they were operating to a certain extent like the training grounds for the new country's professional and political class. Men would make political connections and prove themselves to each other before heading out to shape laws, policies, and religious and cultural norms. The first formal system of documenting merit emerged at West Point in the 1820s and spread to other colleges. This approach to a young man's character - that it could be captured via hash marks, points, checkmarks, etc. - felt fair to the men who benefited from the system. And if a meritocracy was fair to them, it would be fair for their sons.
Before we go into high schools and admission into college, it's helpful to add some more context regarding the diverse higher education landscape in America. Like K-12, higher education is a matter left up to the states, which means 50 different systems, plus the territories and Bureau of Indian Affairs colleges, to say nothing of private colleges, and two-year technical and community colleges. (This comment from /u/gimmebackmyfamily provides more context on land grant and normal [teacher prep] colleges) For most of the 19th century, each college set its own admission criteria and while there were commonalities, it was fairly idiosyncratic based on the nature of the institution. Some Historically Black Colleges and Universities (HBCUs) accepted everyone who applied, even if they were illiterate, with the understanding that basic literacy skills would be part of their college education. Some of the women's colleges (known as the Seven Sisters) only accepted women (mostly white) from particular religious denominations, even if a woman were academically gifted or politically connected. Other colleges, like normal colleges, filtered admission based on the applicants' gender and career aspirations. That is, men who wanted to be school administrators wouldn't go to a teachers' college - and the teacher college wouldn't accept him - no matter how good his character or his grades.
Meanwhile, every generation had a different idea of how to support and achieve meritocracy. Their philosophies would shape evolving admission policies, even at the Colonial Colleges and the colleges founded during the 19th century. As states, towns, churches, and organizations broke ground on new colleges, the networking among the professional and political class continued and allegiances to particular colleges became part of the culture. That is, the notion of a "Yale Man" or "Harvard Man" emerged, which leads us back to high schools and the idea of accreditation and legacy admissions.
Slow fires around the gap between the content and norms of high school and college had been burning since after the Civil War but reached an inferno in the late 1800s. Schoolmen - school administrators, college presidents, and members of state education departments - were determined to solve the problem. Apocryphal stories of "wasted" opportunities became the norm among schoolmen. One college president often told the story that he had to do self-study for an entire year before he could get into the college of his choosing because his high school didn't teach him the right Latin text for the admission exam. College presidents complained about Freshmen showing up, unable to write up to the college's standards. To be honest, they were complaining about the same issue in the 1700s but be that as it may, by the end of the 1800s, there was a concentrated national effort to smooth out some of the wild variations between America's high schools and colleges. And it should be stated explicitly: the same small group of men were part of all of these conversations, most of whom attended Colonial Colleges, most of whom believed America was a meritocracy. So, again, in their view, if a young person applied themselves, checked all of the required academic boxes, and worked hard, they would succeed. Lead among these men was Charlies Eliot, who would appear on nearly every major education committee of the early 1900s. (More on him in a history of social studies in American schools.)
The high school work was primarily led by the National Education Association (NEA) which focused on survey and reporting. Their work informed the creation of the Carnegie Unit, which gave shape to the modern high school liberal arts curriculum and the notion of class periods based on set units of time. The unit wasn't imposed on high schools but rather, on colleges. To shorten a long story, Carnegie wouldn't fund college professors' retirement unless they taught at a "real" college, criteria set by their admission policies. That is, they had to primarily admit students with a high school diploma awarded based on certain criteria. Colleges that participated in the Carnegie program stopped accepting students that did not meet those criteria so schools and districts had to change their schedules and curriculum in order to meet that bar. Colleges could then rest assured that someone who graduated from an accredited high school had the academic part pretty much covered; the college needed to attend to things like "fit" and "character." To be sure, this idea of matching high school exit criteria to college entrance criteria wasn't new. New York State High School Regents diplomas were treated by most colleges as sufficient evidence of readiness for college and although NY's particular system didn't catch on in other states, smaller states often developed their own ways of linking college and high school.
One of the ways to accomplish this linking was the idea of accreditation. This idea was made manifest in associations that provided an early template for athletic conferences and included groups such as the New England Association of Colleges and Secondary Schools, the Middle State Association of Colleges and Secondary Schools, the Southern Association of College and Schools, etc. These Associations acted as a form of self-study that complemented the quantitative measures of the Carnegie Unit and helped give rise to the look of the American high school (despite the lack of a national educational system.) And keep in mind the notion of meritocracy ran through all of this - the associations' goals for high school were based around the notion that any child (by which they were mostly thinking of their and their neighbor's sons) could graduate with high school and be ready for whatever path they chose*, be it college, career, military, or something else.
The College Entrance Examination Board was founded in 1899/1900 and would eventually support the SAT and ACT. However, by that point, states and individual colleges had mostly settled into their particular admission criteria. In many cases, such as NYS, the Regents diploma continued to be more of a marker than a single test score on a test given on a single day. Other colleges, especially those that date back to early America, heavily weigh the notion of legacy. That is, if an applicant's parent (or parents) went to the college, a student's SAT/ACT scores could be virtually meaningless. That said, SAT scores could absolutely be used as a reason to keep a young person out of a particular college, especially as student bodies became incresaingly diverse. Which it to say, Americans - especially those in power or with access to power - may believe in the idea of merit, but it's difficult to say the structures put in place have made it a country where anyone can be successful.
*As an aside, this provides a clear example of how a common argument that American schools were based on a Prussian model aren't on solid ground. Prussian schools evolved into the modern German model where young people attend high schools based on their aptitude and abilities. The norm in America, on the other hand, is for all children to receive a liberal arts education (with some minor exceptions for technical studies) for all thirteen years.
The U.S. system for college admissions evolved as it did primarily due to antisemitism, and a desire to stop Jews from going to elite colleges.
In the United States, selective 4-year colleges and universities typically evaluated students based on primarily academic criteria up until the early 1900s, when this was changed because Jewish applicants were perceived as doing too well on these metrics. The belief of these colleges was that spots in their classes should be reserved for white Anglo-Saxon Protestant men, who often did worse on testing than European Jewish immigrants and their children. Moving towards a holistic notion of “merit” enabled elite colleges to reject Jewish applicants by suggesting Jews, while academically capable, lacked necessary leadership skills and other attributes expected of college students.
Before the move to selective admission
Prior to the efforts to restrict Jews, university admission at elite institutions was determined by objective criteria. African Americans and women were still restricted under most circumstances, but otherwise men were evaluated based primarily on their academic prowess.
Originally universities administered their own admissions tests. In 1916, however, Harvard, Yale, and Princeton moved to a standardized exam administered by the College Board, the ancestor of the modern SAT. Their only other admissions requirement was a satisfactory school record, and exams in “no more than four subjects.” Elite colleges did not even have full offices of admissions, because there was no selection of a class; students either had the numbers to attend or they did not.
By the turn of the century, there was an ideal forming that a college student should be a “well-rounded Christian man.” This was someone of the upper class, athletic, and of Anglo-Saxon Protestant origins. College presidents like A. Lawrence Lowell, Harvard President from 1909 to 1933, feared that Jews, who were largely the children of recent immigrants, would detract from the elite social character they wanted in their colleges.
Jewish quotas and the move to holistic admission
Originally, some colleges considered imposing formal quotas on the number of Jews they admitted, without otherwise changing admissions standards. Lowell, for example, wanted to “state frankly that we thought we could do the most good by not admitting more than a certain percentage of men in a group that did not intermingle with the rest and give our reasons for it to the public.” Overt quotas proved controversial publicly, though, so more subtle ways to limit the number of Jews were found.
Probably the most dramatic change in college admissions was the creation of the nation's first office of admissions at Columbia University in 1910, which was done in response to what they dubbed “the Jewish problem,” by which they meant having large numbers of qualified Jewish applicants. Under the leadership of Adam Leroy Jones, it began to use subjective metrics of “character” and “leadership” to restrict the number of Jews who attended the school. This did not stop all Jews from getting through, and in 1921, Columbia imposed a formal quota which dropped the percentage of Jews attending from 40 percent of the total student population to only 22 percent.
The measures taken by Columbia spread to other schools. By 1922, almost 22 percent of students at Harvard were Jewish, and there were fears these students would drive Anglo-Saxon students from elite boarding schools away. That year, Harvard began requiring applicants to the school to list their race, religion, mother’s maiden name, and submit a form from their high school principal attesting to their religion. By 1924, they were using recommendation letters from teachers and personal interviews to get at an applicant’s “aptitude and character.” The hope at Harvard was to allow Anglo-Saxon Protestant applicants through, while stopping most Jews.
College admissions after Jewish quotas
The idea of “selective admissions,” admissions undertaken by subjective metrics of character rather than scholastic aptitude, continues to be the way college admissions work in the United States. The idea of doing extracurricular activities to get into college is now deeply ingrained in American education.
Yet there were big changes to how selective admissions worked. By the 1960s, American colleges had stopped using subjective standards against Jews and began to use them in a host of other ways to shape the makeup of their classes. Selective admissions are now heavily used, for example, to make college classes more racially diverse, the exact opposite of their original purpose.
Citation:
Jerome Karabel. The Choosen: The Hidden History of Admission and Exclusion at Harvard, Yale, and Princeton. New York, 2005.