Super glue works if it is a clean break without chips. To my knowledge, most (if not all) are not solid pieces of amber, but rather a bonded collection of amber dust or pieces. Who knows what was used as an adhesive back then, but I doubt super glue would be any worse for you, if you're worried about that kind of thing.
Just be sure to put a pipe cleaner in the stem before you glue just in case you're not 100% accurate so you can move it around and ensure the draft stays open. With careful application, you shouldn't have glue in the draft hole anyway.
They're very easy stems to sand if you have a selection of sandpaper up to polishing grit. Should be able to reasonably hide the line. I would try a repair before getting a new stem if it were mine.