Programming is game-like, but agreed, it's more tool-like.
tool: a device or implement, especially one held in the hand, used to carry out a particular function.
Problems that AI can easily solve will hopefully be the more tool-like problems, while humans can think about the more abstract.
You should see my math students who beg me to give them sheets of arithmetic problems.
Play can come in any form of challenge or experience. It's about wether you do it for fun or for a living. A game is more specific and often needs a challenge or a target.
Why is arithmetic not a game? Conway showed the equivalence between numbers and games. Arithmetic procedures are just wrapping a game-like structure around games.