In that case yeah, Oxford probably isn't a good fit as you can only do Hindi as a "minor" language alongside Sanskrit (starting from year 2). I'd agree with you in discounting it as such (at least for undergrad - their Modern South Asian Studies masters course might be of interest later). It would be good if your main interest was Sanskrit but if not, as you say there is somewhat limited return on investment in terms of the topics covered.
Note that for Harvard you are just admitted to Harvard College "in general". You can choose whatever as your major in the end but necessarily about a third of your studies will be across a wide range of subjects (sciences, social sciences, humanities) for general education requirements to earn the degree, and minus the ~half you'll study in your major the rest is free choice. So you have a lot of academic freedom there (and some of it is "imposed freedom" if that makes sense!). This will be the same at Yale.
I'm familiar with LCL at SOAS, it's a new programme and quite broad but unlike Harvard where you can take a broad range of courses that are quite specialist, in LCL you are stuck with a broad range of courses that are usually quite generalist. A lot of the core modules concern e.g. general world literature/history and such and you might only spend a week or two on South Asian topics in that. SOAS currently offers four levels (i.e. years' worth of language modules) of Hindi (and two levels of Urdu) but I don't know if that will continue.
Something you should be aware of is that SOAS has had a LOT of restructuring recently, due to their fairly dire financial situation. Thus a lot of courses got cut, LCL was the result of all the individual regional studies programmes except Middle Eastern Studies and East Asian Studies being cut and rolled into a single degree, and a lot of staff were let go (both academic and non-academic). While at any university it is possible a given module/class may not be offered in future years, at SOAS I would say things are quite tenuous for a lot of more "marginal" languages (i.e. not the really big departments like Chinese/Japanese/Arabic), which unfortunately despite being the cornerstone of SOAS's existence and reputation, are being pretty heavily sidelined due to cost cutting measures. Harvard, having a virtually limitless endowment, is likely to be a lot more stable both academically and administratively.
Overall while I have (and am) enjoying my experiences at SOAS academically, the admin side is a nightmare and it is important to understand the nature of SOAS's situation right now, given your alternatives. Especially if you are getting funded at Harvard, it feels hard to, in good faith, recommend SOAS over it in view of that and the very real possibility you might start your degree and find out next year they're dropping Hindi to 2 language levels (which has been the case for most languages that aren't the big ones) and Urdu might get cut altogether or something...