This is nice. I wonder, though, why people wouldn't have then used some existing, familiar term referring to feudal obligation, instead of "faith".

Because "faith" was the existing familiar term for those relations. The feudal concept of fealty comes from the same Latin root, fidelis, that's translated into English as "faith". The word "faith" is still occasionally used in its original sense, e.g., "faithful servant" or "I have faith in you".