Your observations are correct. does not necessarily imply sexual relations. For example , , and so on are perfectly natural.
(or since is not in the Joyo kanji list) is the one that does imply having sexual relations without getting married. is incorrect.
Of course, or would be actually understood as a proposal for or even marriage in great many cases, especially when this is said between a young man and a woman. I don't know the story of _Fate_ , so I can't judge if it was a bad translation.