I am assuming that you have talked about money for college with your parents. MIT, along with most colleges, have “net price calculators” that will give you a good estimate of your college costs. It will take into account if a sibling is going to college at the same time. If you (or better if your parents do it) have used this, then do so to get a cost. You still might be full pay, but it would be a better estimate than trying to figure it all out yourself.
If your end game is a job in industry, then your performance on the job will dictate your career advancement more than where you went to college.
If academia or research is your end game, then where you went to grad school would mater to some degree. So, GT for your BS and then a “name” college for grad school would work in that case. GT would actually be a good name college.
No matter where you go to college, there are ways to make it at least a bit cheaper. Both of my kids lived in the dorms their freshman year which meant they were also on the college meal plan. After freshman year, they both moved into an off-campus apartment with several roommates. They both knew how to cook and made their own meals for the most part. Doing so saved a decent amount of money. The cost of college was still dominated by the tuition though.
It is good that you are thinking about the cost of college. I’ve seen too many students that are blind to the costs and just take massive loans and are saddled with them seemingly forever.