“I borrow textbooks on the topics, read them, and do every other problem. I use StackExchange and reddit for proof verification whenever the answer isn’t obviously correct.”
One issue with self-study is that it is hard to know how thoroughly you have gotten all the concepts and all the details. It is also hard for a university to judge how well prepared you are. If you sit in a class and take exams, then they can see some concrete evidence of how strong your abilities are.
“I wanted to go to MIT in part because everyone I knew was calling it a ‘math school’.”
MIT is indeed very good for math. I was a math major there. However, there are also a lot of other universities that are very good for math. The top ones (MIT, Harvard, Stanford, …) tend to accept students who they can know are going to do well. Other still very good schools might be more willing to take a chance on a student who has great potential but who took a somewhat unusual path to get where they are now.
There are a substantial number of non-math requirements to get a bachelor’s degree at MIT.
“Nearly all of my bad grades were freshman year”
Freshman year will matter a lot less than more recent years.
“I hope that college will be better.”
I think that it will be. Even if you go to a “pretty good” university (such as an in-state public university in an average state) there will be professors who know their subjects very well, and there will be a LOT more students like you than there were in high school. If there are subjects that you really like, you can in university visit the professor and get extra work in the subject – I know that this seems like a strange thing to do but it can be and sometimes is done.
If you go to a “top 100” university for undergrad and do very well, this will open up the possibility of going to a “top 10 or maybe even top 3” university for graduate school.