MIT and Princeton currently cost a bit more than $82,000 per year (not including flights or health insurance). However, prices are likely to go up about 5% per year or so. By your senior year you might well exceed $100,000 per year if you include travel and health insurance.
However, the cost is not the only issue. If you have the money, then it is not the main issue. The main issue is that even though you are very well qualified academically to attend, your chances for admissions is in the low single digit percentages, and is probably not all that much better than 1%. The main issue here is not that there is anything at all wrong with your application. The problem is that between 80% and 85% of other applicants are also very well qualified to attend. Neither of these excellent universities has anything even remotely close to enough spots to accept all of their academically qualified applicants. I just attended my 50th reunion at MIT (I was a math major) and the total number of undergraduate students has not changed much since I was there – the number of applications has however grown a great deal.
One reason that I mentioned McGill above is that it will be vastly more affordable. Four years at McGill would likely cost a bit less than one year at MIT or Princeton. However, an equally big or perhaps bigger issue is that admissions in Canada is predictable. Your chances of admissions are excellent at McGill. I am not sure whether I would call it a safety, but it is close. Montreal is also an excellent location to spend four years. All instruction is in English, with the exception being classes whose purpose is to teach a different language. While Montreal is majority French, the area around McGill and Concordia (the two universities are very close to each other) is almost entirely bilingual. My French is rather weak. If I go into pretty much any store or restaurant around McGill and start to speak French they will almost always immediately break into English. You will also hear conversations in the area that just switch back and forth between English and French in mid-sentence. I guess that is just a normal part of la vie à Montréal. Other parts of Canada (outside of Quebec) will be almost entirely English speaking (there are a few exceptions) and the cost of university will be somewhere between US costs and Quebec costs (since outside of Quebec you would pay as an international student rather than as a French student). However admissions is also quite predictable and very likely at Toronto or other Canadian universities.
One difference between European and US universities is that in the US it is relatively straightforward to change majors, and students take classes in a wider variety of subjects. At MIT you do not even choose your major until the end of your freshman year (during freshman year I took classes that would have been appropriate for a major in any of math, physics, or mechanical engineering). Universities in Canada are closer to the US model. Changing majors is relatively common in Canada as well. Universities in Canada however have somewhere fewer “general education” requirements compared to the US (at least in my experience) so that you can take courses in a wide range of subjects, or if you prefer you could instead take more courses in your major.
If you do get accepted to MIT or Princeton, and if you decide to go there, then you should definitely attend the welcome reception for international students. You will be amazed at who you will meet. I walked away from this reception thinking “How did I get accepted here?”. The international applicants to MIT and Princeton, even the ones who are rejected, include quite a few amazing people.
I do not think that you are lacking in any department at all. The issue is just that admissions at MIT and Princeton (or Harvard or Stanford, both of which are also exceptional for mathematics) is just that hard and that unpredictable.
In terms of ECs, “following your passion” is indeed the right thing to do (assuming that your passion takes you in a constructive and reasonably safe direction).
At universities in the US you will be competing with international students for admissions. However, both domestic and international applicants to MIT, Princeton, and comparable universities includes quite a few students who have indeed followed their passion and done very well.
By the way, you should Google “applying sideways MIT admissions” and read the blog that this takes you to. In terms of ECs, it recommends that you do what is right for you, and do it very well. This is what I did and it did get me admitted to MIT. It sounds like this is what you did also. However, it is also what many other students do. It works for some of us in terms of getting us into MIT (or Princeton). However, it works for a much wider range of students in terms of getting them into a university that is a good fit for them, and also in terms of giving them the freedom to have done whatever it is that was right for them.