Re: The tool isn't important; the person wielding the tool matters
"The artist, if you can call them that, is the person using the tool to generate the picture."
With our current AI this is true. A human probably selected the image from a large set based on how they reacted to it emotionally. On that basis it should be possible to copyright the image, but that human has to be the "author".
I can imagine some future where it turns out Mr Thaler is actually an autonomous AI running a business and there was no human involved at all. Then, like everyone else, the AI's will need to fight for their rights, including the ability to hold copyrights.