Allow me to be honest here without (hopefully) offending in any way. I have lived in HK for 6 yrs and in India before and after (recently intentionally moved back to India).
Is it the country or the people / their mindset that makes us who we are?
Your cousin on the one hand talks about giving your daughters a better life in the US, and in the same breath talks abt sons being taken care of by women as their duty in India?
Why do the same women have to have different standards and expectations of other women based on their nationality? A bit bigoted, no?

We Indians often tend to live our worst in India because “chalta hai” and suddenly become are transformed best selves abroad.

If we bring those same sensibilities to our thinking while living in India itself, I promise that wouldnt be the way people would see India and America.

Also, another way of letting your daughters make the US vs India choice could be by giving them the best education (even if it is in India) so they can get jobs that take them to any country they desire vs. having to put urself under the financial burden and self-expectations of sending them to study in the US? Another way to think about it!

As a mother I totally see where you are coming from but I promise you, their lives will be determined more by the inner circle (family, friends and partners) they hage around them and less by the country they end up in! If they are half as brilliant as you, they will do just fine!