Yes, Schfifty Five is correct. You are correct in your conclusion, Nik, but incorrect as to your substitute reason why, and completely incorrect in saying Shfifty is incorrect.
The reason that "I" is correct is because it is the subject of the second, unspoken clause. Verbosity, or the lack of it, has nothing to do with it.
It is not verbose at all to say, "You run faster than I do" or "You run faster than I run" at all, it is merely also correct in English to just say, "You run faster than I."
Search the entire corpus of English grammar on this subject, and you will not find one instance of verbosity being mentioned as pertaining to this usage question in any way, shape or form.
Otoh, you will find that every single grammarian does say that "I" is correct because it is the subject of the hidden or truncated second phrase, as Shfifty said was the case.
Peace. We are all wrong at times. :thumbsup: