Some phrases just sound odd or dated to the modern ear if you just stop to think about them. The phrase "carrot and stick approach" is one such example.
The phrase's origins were to describe getting a horse or a donkey to go where you wanted by holding out a carrot in front of them as a reward or threatening to beat them with a stick if the carrot did not work. To begin with - horses and donkeys are not very prevalent in today's society and many people in today's generation don't like carrots so probably don't get why carrots are chosen as a symbol of a reward. To them saying carrots is like saying "if you are good I'll give you extra broccoli with your dinner". Where's the reward in that?
And the stick? "Do what I want you to do or I will beat you with a stick!" That's so many levels of wrong in today's society - doesn't make a difference if you are talking man or beast. Just threatening someone with a stick is legally assault. Actually doing it to a horse or donkey would get the ASPCA all up in your grill.
Maybe it's past time to drop the phrase "carrot and stick" from the vernacular.