Chances as an international student seeking financial aid?!?

Hello everyone,
So Im Greek and I really want to study in the US. Problem is my family is kinda broke so I will basically need a full ride in order to attend any college.
That being said, after some research I realised that the only colleges that cover full tuition for international students are either Ivies or extremely selective universities (below 20% acceptance rates). And most of them consider your ability to pay, which means they
d choose a less qualified applicant that can afford to pay over me.

So yeah… things are pretty tough.
Id really appreciate any advice and your opinions on whether i have a shot at the Ivies and what not.
Here
s a brief description of myself:

High school grades: stellar
SAT: 2200
Took SAT II Math level 2 and Physics, both will probably be above 750
Im a scout
I participated in an entrepreneurship club where my team created 2 companies, and i was the CEO of one of them, got national awards and all that jazz
I founded a club at my school to encourage teenagers to start volunteering (volunteering isn
t really big here in Greece)
finalist last year in the national informatics competition
Placed 2nd in my grade in the AMC 12 competition
Recently joined Mensa
Play the piano and the guitar and am part of a school rock band
Member of various theater clubs since i was 7
Worked for 4 summers in a pharmacy
Play tennis as a hobby, not anything major

Other random info im not sure will affect my chances:
i attend a private school with a full-tuition scholarship
lost one of my parents in elementary school (will be mentioned in commonapp essay)
undecided major, but something within the STEM fields
i
m a girl

Any advice is appreciated! Remember, we`re talking about the elite schools here (Yale, Princeton, UChicago, Johns Hopkins etc.)

I’m in rhe same situation as you are,but I am from Romania.I think you should improve your SAT score and the rest is pretty good.I can’t tell if you will get in tho,but good luck!

Thanks! You too! I retook my SAT and got a 2290, so I`m good in that department. Now all I can do is polish those essays and hope for some good luck. After all, Ivies are a reach for everyone. Good luck on your applications!