Hi! I can speak specifically to doing an MPhil at Oxbridge but not to American PhDs. I'm currently doing an MPhil in IR at Oxbridge, and based on your specific research interests, Cambridge especially sounds like a great fit for you in terms of coursework but Oxford is slightly better with social sciences and has a more critical curriculum (I applied to and researched both programs carefully). Congratulations on being accepted and best of luck with all your other applications!
Probably one of the best parts about doing an Oxbridge MPhil is that MPhils are considered to be between an MA and PhD (since they're so research intensive, even with mixed taught-research programs), so I believe it would definitely give you an advantage later when applying for PhDs. Another thing to consider (and I don't believe this is fair, but just the way it is and many people independently have confirmed this to me during my studies) is that having a master's from an Ivy League (the obvious ones in the US + Oxbridge) will give you a definite advantage when later applying for professorships, especially in academia that is already so competitive for jobs. I think it's always better to diversify your academic experience (i.e. undergrad from India, master's from the UK, PhD from the US).
Another thing to consider is that in my experience grading is way harsher at Oxbridge (basically you only get distinction if your paper is publishable in peer review) HOWEVER I've often been told this is taken into consideration when applying to non-Oxbridge PhDs and will still give you the advantage over someone with a higher average but from a non-Oxbridge-level school. This is just my experience, and having said all of this I firmly believe you should make a decision based not only on what will get you to your eventual goal of PhD / teaching, but also on which program specifically is the right fit for you and where you think you'll be the happiest! Best of luck and please don't hesitate to message me if you have any other questions I can answer.