Earlier in US history, it was common for universities to admit students via entrance exam. A quote in this answer quoting Colonial Education explains "The boys from upper class families were taught be private home tutors and then sent to college or university."
Nowadays, a high school diploma or an officially recognized equivalency (such as passing the GED exam) is generally recognized as the standard way to qualify to start studying at a US institution of higher education. Although such a credential is not strictly an absolute requirement even today, the idea of "finish high school, go to college" seems to have been standard in the USA at least as far back as the 1950's (with enrollment of "non-completers" or "dropouts" as the exception to the rule rather than standard practice), as my parents reported to me that essentially the same formal expectations existed then.
At what point did graduation from a "High School" or the achievement of an equivalency qualification specifically recognized as equivalent to high school become the standard way to get into a university, with other paths becoming the exception rather than the rule?
Yes, I know that it is still possible for someone to be homeschooled, pass the GED at age 18, and then go straightaway to college, but the GED is specifically recognized as high school equivalent, not a truly distinct academic pathway. I'm talking about a time period when there was no concept of a high school equivalency, or when the achievement of a high school qualification was truly an optional part of the process of qualifying for university. That is, an aspiring university student might have chosen to complete a high school qualification because their parents wanted them to, because they wanted the "high school experience", because they were being funded to attend, because it was easy for them, etc. but they could have reasonably and without too much head-turning short-circuited the process and/or followed an older, still common alternative path to a university (e.g. private tutoring toward passing a direct entrance examination, getting sponsorship from a local politician, making a
substantial bribe large grant offering, etc.) if they had wanted to.
Alternately, when was the first time in US history where an incoming Freshman might hear something like, "Wow, you did the thing where you study at home with private tutors and then get five alumni to endorse your academic abilities? What a weirdo! Nobody does that anymore! If you want to be in the "cool" crowd, it's all about getting your high school diploma! I wouldn't be surprised if the university abolishes all those non-high school paths next year."