If you don't get an answer here, you might want to try /r/cfb. It's obviously not a historical sub but there are a decent number of people on there who are very knowledgeable on the history of college football, and especially those smaller/obscure teams, so you might be able to get an answer if no one here can provide one.

Oh my lord, finally a question I can answer!

Prior to 1937, St. John's College (henceforth the College) was on the brink of losing it's accreditation and generally in fairly dire financial straits. A new president was appointed, by the unfortunate name of Stringfellow Barr. Barr was a bit of an educational radical, influenced strongly by both his position on the faculty at University of Virginia, and meetings with fellow academics in Chicago.

Barr came in announcing that the College would be undergoing a completely radical shift, intended to turn the College into a sort of academic Shangri-La. Over the next two years, he implemented an all-compulsory curriculum based on the "Great Books", taught through seminar-style discussion classes. This "New Program", as he called it, included foreign language (Latin, Greek, French, and Russian), mathematics, philosophy, theology, and laboratory sciences. Barr further did away with all academic departments and tenure; all faculty members were expected to teach across the curriculum and were henceforth referred to as "tutors" instead of professors.

As part of this transformation, Barr did away with all intercollegiate sports, and replaced them with intramural teams. His vision was for all aspects of the College to be accessible to the entire polity or community, and that included athletics. Athletics were there to complement the Program, and were specifically not to be a self-supporting facet of the school.

Since then, St. John's has retained essentially the same vision as Barr laid out. The College is now larger (there is a second campus in Santa Fe, NM), and also offers two graduate degrees (one in liberal arts and the other in eastern classics). There have been some changes to the Program (for example, Latin and Russian are no longer taught; the language component is two years of Greek and French each), but the idea of an academics-first institution with a curriculum built on a selection of 'Great Books' remains. There are still intramural sports, and occasional friendly competition with other schools.

However, since 1983, the College has engaged in one regular intercollegiate competition. That year, a midshipman at the College's neighboring institution, the US Naval Academy, mused that Midshipmen could easily wipe the floor with Johnnies at any sport. A clever Johnny proposed a croquet matchup, and a long-running tradition was born. The USNA allegedly hired a professional croquet coach as of 2006, but despite the US Navy's best efforts, the rag-tag team of Johnnies has won 30 of the 37 years the competition has been going on.

Sources: "In the Beginning...The Genesis of the St. John's Program, 1937", Peter Nelson, The St. John's Review, Vol. 49, No. 1, 2006. https://issuu.com/stjohnscollege/docs/sjc_review_vol49_no1

Stringfellow Barr, Educator, Dies; Pressed Study of 100 Great Books. David W. Dunlap. New York Times, 5 Feb 1982. https://www.nytimes.com/1982/02/05/obituaries/stringfellow-barr-educator-dies-pressed-study-of-100-great-books.html

Goliath Braces for David, Part 24 Naval Academy Aims to Upend St. John's in Croquet Rivalry. Daniel de Vise. Washington Post, 23 April 2006. https://www.washingtonpost.com/archive/politics/2006/04/23/goliath-braces-for-david-part-24-span-classbankheadnaval-academy-aims-to-upend-st-johns-in-croquet-rivalryspan/711c2e01-2a2c-4aab-9004-5e9180f836f7/

Croquet Fact Sheet. St. John's College. https://www.sjc.edu/annapolis/events/croquet/facts