Homer is denied entry from several colleges before going to Springfield University thanks to Mr. Burns pulling strings. In reality almost all colleges and universities in the United States will just automatically enroll adults, typically ages 22 and over, into a Bachelor's degree program without the application process that recent high school students go through (SAT, ACT, application essay, letters of recommendation, etc). Homer could've just gone to the Admission's office directly and enrolled automatically into college.
When Mr. Burns gets Homer into college, Homer says "I won't be needing my high school diploma anymore!" But Homer has a GED, not a high school diploma (and the document he sets fire to even clearly has GED written on it.)