To be clear,
pressing a button is not copyrightable.
The decision about where to take a picture, what settings to use, and how to take it is the creativity that grants the image copyright. There are various arguments about which parts of prompt generation and refinement might register as copyright and there are various arguments about how tuning settings and tuning prompts is similarly creative to using a camera, but the point is still that unless the user input is sufficiently creative, the image wouldn't get copyright. The button doesn't matter; the button is not what gives you copyright.
In fact, numerous accidental photos have been denied copyright; most famously when PETA sued a photographer over a picture that was taken when a monkey stole the photographer's camera. The court's decision was that nobody owned the photo. It was an accident, it wasn't the result of a creative decision. There was not enough human creativity involved in the process of a monkey stealing a camera to warrant protection.
Of course, it's somewhat of an oversimplification of prompt engineering to phrase it as just saying "elephant with blue skin", but if that was the entire creative input, it's not clear at all to me that someone saying "I want an image of an elephant with a blue skin" is a sufficiently creative input that it should be copyrightable. What the AI does with that prompt is irrelevant, it's the human creativity that matters. Same with photography; the camera isn't really the important part. The machine isn't what is generating the copyright. The person making a conscious decision about where to stand, what settings to use, and when to press the button is viewed by the law as a creative act that requires creative skill and execution. The button press itself doesn't matter.
Again, to be clear, prompt generation tends to be oversimplified in these conversations and I'm not saying there's definitely nothing creative happening, but if we take that simplified version of prompt generation at face value, then just saying what image you want... does that really meet a creative standard?
Because saying that descriptions/requests on their own are sufficiently creative for copyright protection has implications far beyond AI art; it implies that commissioning a piece of artwork even from a human being should be enough of a creative act that I should get joint copyright over the final image. If the argument is that prompt generation is more than that, and that it takes more skill, then fine -- but if the argument is that even just a one line description of what image you want should be counted as creative... yeah, that's a pretty significant expansion of copyright that will affect a lot more than just AI art.
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Edit: I think people also get a little confused about the difference between how people generally treat photography and what the law would decide if the copyright on every single image was challenged.
If I set up a camera mounted to a pole, focused on a static scene, and you walk over and press the button on the camera, legally you very likely don't have any copyright over that image. But practically, nobody is going to challenge you over it.
It's possible that some of the photographs where people say, "well that gets copyright, why doesn't mine?" might not actually get copyright if they were ever challenged. But people generally don't challenge copyright status in the first place. Recipes, APIs, monkeys, (and apparently now AI images) are the rare exceptions.