If I set up an entire scene with props and artwork for a photoshoot with a model, but I would like to actually be the model so I ask a friend to go behind the tripod and tap the shutter, the friend holds the copyright?
The monkey situation is kinda screwy of course because ... a contract with a monkey?
"No copyright" as a result in the monkey case seems like a technically legally correct but sad outcome.
For AI tools it's likely currently technically correct as well, but the law probably needs some updating.
there's a scene in one of those Matthew McConaughey romcoms where he plays a photog. The crew has a scene completely setup up and ready to go so that he just walks in, hits the shutter release one time, and then walks away with little care as job is done. He's now credited for that photo, yet did the least effort possible. (that scene isn't too far off while only slightly hyperbolic)
It seems almost directly analogous to asking the AI for an image that you imagine.
Did they have any creative input before hitting the shutter? Did they tell you to move? Or adjust lighting? Did they choose the angle or framing?
These answers influence the answer to your question.