Again, your understanding is…imperfect.
I’m not sure what is included in “like me”- but the parts you put here- smart / hard-working / high achieving / ambitious with a strong public service focus- are indeed the sort of people they want. If it’s the ‘impoverished background’ or the first generation US citizen part you think they don’t want, you are wrong about that as well!
The challenge with the super-selectives isn’t that they won’t want you- it’s that the sheer numbers of super-qualified applicants means that there simply aren’t enough space for everybody. Do your stats put you in the cohort of your class that is likely to get offers from Ivys/MIT/etc?
You haven’t been specific about this, but you alluded to your family’s financial situation being ‘complex’. Does that mean that the net-price calculators (NPC) yield numbers that are not affordable? Even (say) Cornell? Georgetown?
Fwiw, stipends for public-service oriented internships exist at a lot of colleges and universities- typically $3-5K/ for a summer. The NSF has some REUs in the Social Sciences, and some of policy places stipend (though many try to weasel by giving ‘academic credit’, which doesn’t pay the rent). If you have EU citizenship that opens even more options.
One piece that may not be obvious is that the better you can figure out what you are looking for the better you can target schools. Remember that whatever you start out thinking can evolve and grow- nobody expects that how you see your path at 18 will be the same as at 21 / 25 / 30!- but it will help you start building expertise in an area of genuine interest. For early-stage career that is massively helpful.
So going on the bits you mention in your posts above, you seem interested in the intersection of (environmental?) science, public policy and IA/IR. Do you find things like these interesting?
https://www.internationalaffairshouse.org/environmentalism-in-ir-theory/
https://duckofminerva.com/2017/04/why-ir-needs-the-environment-and-the-environment-needs-ir.html
Even if it’s not environmental science, having actual science credentials is an advantage in policy circles- because many students divide along the quant/qual humanities/science axis you mention above. A student who is both comfortable and credentialed at the interface stands out. If Georgetown isn’t affordable there are other schools where you can do a combined science + IA/IR degree.
Finally, are you familiar with natsecgirlsquad.com? Although it is focused on the national security and defense end of things, it is a super resource for young women looking for careers across the public policy spectrum. My public policy collegekid found her current (Cairo based, but virtual) internship through their listserv.