It isn’t just that these schools are need-blind. Their financial aid policies do not include loans and are generous. Students whose families make under $65k go almost for free, and those making under $150k pay 10%. This aid is the same for internationals and there may also be travel allowances.
https://college.harvard.edu/admissions/apply/international-applicants
The residential house system breaks Harvard up into smaller communities, which is a great feature of the experience. Each house has its own dining hall, where people may congregate, its own events and staff.
Your socioeconomic background may even be a plus, if it means you overcame certain obstacles to achievement.
But international admissions are low and it would be wise to have other “dream schools.” Amherst and Williams are also know for good FA.