How did CollegeBoard manage to get its SAT and AP tests to be such important steps in getting into most universities in the US?

The short answer to your question is that the College Board is the Old Boys Club in education made tangible.

The College Board (originally called the College Entrance Examination Board) is basically one of the first educational think-tanks in this country, founded in 1899 by educational leaders including Nicholas Murray Butler, future president of Columbia, with the explicit purpose of strengthening the connection between secondary and tertiary education. The Board would be joined in the educational landscape by others including The Carnegie Foundation, The Ford Foundation, and American Council on Education, all focused on "improving" education. Due the early leadership of Henry Chauncey, the Board gave some of their tests over to Educational Testing Services (ETS) and became a major player in the assessment of Americans - when scientific management became all the rage for schoolmen, they advocated for the new invention that was the multiple choice test. As public high schools advocated for courses for their students that would earn them college credit, similar to what some private schools did, the Board made Advanced Placement courses available to more schools. And when the gap between college acceptance criteria and high school exit skills became noticeably large, they were instrumental in making the SAT part of school culture.

AP has been around since the late 1950's when it was started by a group of private schools in partnership with Harvard, Princeton, and Yale. In effect, the AP exams were about giving young men from families of means an extra leg up on their path to college. AP would expand beyond private schools and Ivy League colleges and is now a way for High School students to earn college credit. In some cases, high schools give extra weight to AP courses on a student's HS transcript, which can elevate their Grade Point Average (GPA). In addition, college admissions offices may view students who've successfully completed AP courses more favorably as it suggests they can handle college-level reading demands and rigor.

The College Board not only shepherded the Advanced Placement exams, they were also part of the evolution of the mental tests like the Army Alpha test, IQ tests, and the GRE. Chauncey, who was responsible for the College Boards' networking prowess and eventually the founding of Educational Testing Services (ETS) was deeply committed to the idea of meritocracy. Informed by the work of men like Alfred Binet and Edward Thorndike, he believed it was possible to create a test that would measure a learner's true ability and that test could help make America a true meritocracy.

He wasn't alone in this sentiment. American public schools began to adopt a philosophy of increasingly secular, liberal arts in the mid-1800's. The general goal was to provide all children (all never means all, though - mostly white children, mostly boys) could best be prepared for citizenship through a broad curriculum that touched on a variety of topics in a variety of ways, without requiring a young person commit to any particular future. This vision contrasted with the then reality of college - most East Coast colleges and universities had their own admissions exams, often very specific to the professors who worked there. By the late-1800's, the gap between college admissions criteria and high school exit skills was becoming an explicit source of frustration for schoolmen. One of the clearer examples of this involved a man who would eventually become a college president. He wanted to attend a particular university but when he arrived for the entrance exam and interview, realized his high school curriculum didn't align to what was expected and he had to spend at least a year studying with a tutor before taking their particular entrance exam. His personal experiences would be a recurring talking point at educational conferences in the early 1900's and is a compelling example of how personal a great deal of educational reform has been. This gap explicitly challenged the notion that America was a place where anyone (mostly white, mostly boys) could succeed, provided they worked hard enough.

The evolution of the SATs was the result of relationships between eugenicists and educators and there was a feedback loop between colleges, the US Army, and high schools as they developed, what they felt, were better and better tools for assessing things that were previously unknowable. Most of the leaders involved in the College Board's work were exposed to formal education from childhood, were from families with the means to provide their son with advanced education, and were white. None of the them were women. College presidents, private school headmasters, district superintendents, and leaders of groups like the College Board generally came from the same pool of East Coast educated of men, meaning those who advocated for change were typically viewing that change through a particular lens and pool of experience. That the College Board members' philosophy that tests can distinguish levels of thinking and abilities is so widely accepted today speaks to the networking that happened.

Further Reading: This is a pretty solid timeline of the SAT's evolution over time

Lemann, N. (2000). The big test: The secret history of the American meritocracy. Macmillan.

Riccards, M. (2009). College Board and American Higher Education, Fairleigh Dickinson University Press.