I can do that without a graphic design school or program! Is there anything else that I’d be missing?

Well, a load of potentially strong connections for one. Your classes would be filled with like-minded people that are interested in a lot of the same things as you. Most often, schools will have connections in one way or another with the industry. Some employers almost have a partnership with some of these graphic design schools.
You will most likely miss some of the teamwork aspects of things, which can be really important if you end up at a graphic design studio. Learning to work along side with others can be really important. This is most often a part of the graphic design program.
You can ask your educators or classmates questions about anything you are unsure about. You can get feedback and critiques, learn and grow a lot faster in a class environment. In a solo environment, you have no real way to measure how good or bad you are, and if you are on track or missing the mark.
Most good programs offer a work practicum, which is a great way to get your feet wet with some experience, and at the same time, potentially land you a job where you’ve been placed.