A histogram cannot output similar images, it's pointless to argue the fine details of an analogy that doesn't apply to begin with

To call it "stealing" might be inaccurate, but are the artists wrong to say that their intellectual property rights are being violated, when people using their works without consent to train AIs with the express purpose of replicating those artists' works? I have seen several artists pointing out AI users who brag to them that they are explicitly training AIs using those artists' galleries and show that it's outputting similar works.

The reason why Copyright even exists, at least ideally, is so that the rights and livelihood of artists is protected and they are incentivized to continue creating.

Case in point. That's not why copyright exists. The reason for the American version of copyright is established right in the constitution: "To promote the progress of science and useful arts".

How is it "promoting the progress of useful arts" not the same as "incentivizing artists to continue creating"? Are you going to argue what's "useful"? If there is interest in replicating artists' styles with AI, then that is an admission the people doing it see use in those works. Otherwise, it's the same, and protecting their livelihoods through the privilege of a temporary intellectual monopoly is how that promotion of arts is done.

I definitely see the value of the Public Domain, but if expanding it at any cost was the primary goal of copyright we wouldn't have roughly century-long copyright. Which I don't think is good per see but that's another discussion. Still, the existence of copyright at all is a concession that grants that for artists and creators to develop their works and ultimately enrich humanity's culture, they need to be able to control their works and have a guarantee to a stable career, to the extent that they can sell their own work. It's a protection so that not everyone can show up imitating that artist and undercut them, undermining their capability to make new creative works. Which is what many people have been doing with AI.

If anything that could enrich the Public Domain was enough reason to drop Copyright, we wouldn't have any Copyright. The compromise is that Public Domain as a whole will be enriched when the artist's Copyright expires.