College education was intended for the elite. The proportion of the population that possessed a high school diploma, much less a college degree was infinitesimal. Indeed, as late as the 1950s, only ~8% of the population were college graduates.

In practice, the requirement for the classics was no impediment to elites; one could be admitted on the condition that shortcomings would be remediated. Once admitted, you could easily purchase the services of a tutor or cheat your way through.

The European university tradition on which Harvard was created treated Latin and Greek as the languages of learning. At the time a college education included a rigorous treatment of the classics (Latin) and philosophy (Greek). The belief was probably that they couldn't be truly appreciated understood unless studied in the original language (which is arguably true). And of practical note, some scientific writing was still done in Latin at the time and the language was still in wide use in throughout Europe.

Heck, I learned Latin in high school in the late 90s. I hated it when I took it, but it certainly made learning other things a lot easier.