Sorry to be insistent, but BERUF is the word used in my German A1 book... and I would argue there were some professions (most of them unusual, I grant you that) which weren't a craft, in the sense of making something with your hands: politician, priest, vestal virgin, taster, accountant, author...
Do you still disagree?
„What do you do?“ implies a broader choice of occupation (e.g. student) while „What is your profession?“ implies you know (or think that you know) this person has a profession and you ask what they do for a living. „What do you do?“ can mean the same thing but only with proper context, which there isn't.
Yes, "career" is "die Karriere", and implies what you do your whole life (places of employment, specific activities there and so on) and how it goes, more than a "profession" you may even never have a chance to practice after training for it (I know of numerous exemples).
For instance, an actor's profession is "actor". Their career is: What movies, what plays, what series did they do, what awards did they won, did they also direct, wrote books, sang, dance, etc.
I think my translation "what is your occupation" should be accepted. Occupation is more general, and includes professionals. How many applications ask what your profession is...it's always "what is your occupation". A doctor or a wrestler could both call themselves professionals? why so picky?