The only 'problem' of preposition stranding is that it is a unique feature of English that doesn't exist in Latin, French or Standard German. It's something that learners of English as a foreign language get drills for because without doing it correctly you can't speak proper, natural, effective English.
But of course English prescriptivism is mostly isolated from the actual facts of its grammar and from how the best authors use it. It's primarily about setting up artificial standards whose purpose is for them to be hard to follow even for native speakers with excellent instincts and a large input of excellent literature. They operate as shibboleths, or secret handshakes that allow people who have gone through the same objectively nonsensical training to recognise each other so they can dismiss everyone else. This is why distances such as 5 miles magically become countable rather than approximations when it can serve as an excuse to trash people who use the perfectly natural (and only correct) less than five miles, and this is why not even Fowler was able to stop the "cherished superstition" that there is something wrong with preposition stranding. (It's normal that prescriptivists ignore the unanimous opinion of all linguists, but given that most pay at least lip service to Fowler, one would expect his opinion to carry some weight.)
I wasn't aware it was accepted! In that case, I'll also offer my opinion on this. ^_^
"To which hospital do you go?" is technically more proper—it's more formal, meaning you would want to use it in more academic settings, for example. However, on a day-to-day basis I really believe the default sentence is more correct, as it's what's more natural to native speakers. Truthfully, I'm not entirely fond of the rule stating that prepositions can't go at the end of sentences—though in certain cases I do think it sounds better—because that rule was proposed by someone who said that English shouldn't allow prepositions at the end of sentences because that's what Latin did. English and Latin are, however, very different languages. At the end of the day, what native speakers use is what's technically correct, and since languages naturally evolve through time I don't think people should necessarily try to stop that. As time goes on, it's possible putting prepositions towards the front of clauses will sound downright strange. :o
Bear in mind I'm not disagreeing with what you said—I'm simply stating my personal opinion. Grammar can make for interesting discussions. ^_^
I ceased about 40 years ago when I read an article published around 1923 called "Preposition at end", which included the following dialogue: Sick Child. I want to be read to. Nurse. What book do you want to be read to out of? S. C. Robinson Crusoe. (Nurse goes out and returns with 'The Swiss Family Robinson'.) S. C. What did you bring me that book to be read to out of for! *** Eschewing modern usage will however cause awkward sentences that can be used with tongue in cheek, such as "Up whom are you looking?" to someone searching for a number in the telephone book.