The term carrot and stick doesn't sit well with me. If you hit a donkey with a stick then it's likely it won't trust you enough to accept a carrot later.

its an archaism. a modern understanding of animal behavior and ethics have both pushed that training modality out, and probably the more successful animal trainers of yore never really relied on it, but people absolutely used to beat the shit out of animals. perhaps not often, and perhaps not horribly, but animal cruelty was certainly common enough to enter the lexicon. see "carriage whip". see "spurs". see "bearing rein". see "branding iron". etc etc. hell theres even the phrase "spare the rod, spoil the child".

and this is not the worst place a study of the history of language will take you. remember that european and white north american cultures both (among others) unabashedly practiced chattel slavery for centuries, if not millennia for older cultures.

This is a little naiive imho. I get what you're saying, but the reality that has been proven time and time again is that if you're willing to stoop low enough and cause enough suffering, it is possible to break most animals' spirits.

In addition to all of the other answers, I've heard the phrase "carrot or the stick" significantly more than "and"

I think it's more like a threat

On the contrary, that's why it's perfect. There is so much evidence in so many different context that positive reinforcement yields better results than negative.

This is congruent with your donkey example.

No you're not hitting it with enough force or frequency with the stick if it's not working. You have to entirely break an animal's spirit and then you can encourage it. Circuses have been doing this for centuries