Over my life I have visited the flagship campuses in Arizona, Indiana, Iowa, Maryland, Massachusetts, Michigan, Minnesota, Nebraska, New Jersey (Rutgers), North Carolina, Oregon, Pennsylvania (Penn State), South Carolina, Texas, Vermont, Virginia, Washington, West Virginia and Wisconsin. (Illinois is conspicuously absent). I have also visited several campuses of the California and New York systems which don't have a specific flagship. I can more locations of flagship campuses than I can state capitals.
The state universities got a strong push from the Morrill Land Grant Act passed during the Lincoln administration after the southern states that objected (like South Carolina) had left the union.
One of the great strengths of the US higher education system are these state universities, independent and competing, instead of a system of nationalized universities that you find in many other countries.