All of the great presidents came from no-name colleges (Ronald Reagan, Thomas Jefferson, James Monroe) or didn't even attend college at all (George Washington, Abraham Lincoln, Grover Cleveland).
Dwight Eisenhower was one of our greats, but he was trashed continually by the left. He went to West Point.
Perhaps this is a Virginia-born bias, but I wouldn’t exactly call the College of William & Mary a non-name college.