It doesn’t have to have a copy of all copyrighted works it trained from in order to violate copyright law, just a single one.

Sure, which would create liability to that one work's copyright owner; not to every author. Each violation has to be independently shown: it's not enough to say "well, it recited Harry Potter so therefore it knows Star Wars too;" it has to be separately shown to recite Star Wars.

It's not surprising that some works can be recited; just as it's not surprising for a person to remember the full text of some poem they read in school. However, it would be very surprising if all works from the training data can be recited this way, just as it's surprising if someone remembers every poem they ever read.

how easily accessible does a copy of a copyrighted work have to be from an otherwise openly accessible data store in order to violate copyright?

I don't think it really matters how accessible it is, what matters is the purpose of use. In a nutshell, fair use covers education, news and criticism. After that, the first consideration is whether the use is commercial in nature.

ChatGPT's use isn't education (research), they're developing a commercial product - even the early versions were not so much prototypes but a part of the same product they have today. Even if it were considered as a research fair use exception, the product absolutely is commercial in nature.

Whether or not data was openly accessible doesn't really matter - more than likely the accessible data itself is a copyright violation. That would be a separate violation, but it absolutely does not excuse ChatGPT's subsequent violation. ChatGPT also isn't just reading the data at its source, it's copying it into its training dataset, and that copying is unlicensed.

information that it can index, tag and sort for keywords.

The dataset ChatGPT uses to train on contains data copied unlawfully. They're not just reading the data at its source, they're copying the data into a training database without sufficient license.

Whether ChatGPT itself contains all the works is debatable - is it just word relationships when the system can reproduce significant chunks of copyrighted data from those relationships? - but the process of training inherently requires unlicensed copying.

In terms of fair use, they could argue a research exemption, but this isn't really research, it's product development. The database isn't available as part of scientific research, it's protected as a trade secret. Even if it was considered research, it absolutely is commercial in nature.

In my opinion, there is a stronger argument that OpenAI have broken copyright for commercial gain than that they are legitimately performing fair use copying for the benefit of society.

I'm honestly not sure what you're trying to say here. If by "it must have access to information for reference" you mean it has access while it is running, it doesn't. Like I said that information is only available during training. Either you're trying to make a point I'm just not getting or you are misunderstanding how neural networks function.

I think what they mean is that ML models generally don't directly store their training data, but that they instead use it to form a compressed latent space. Some elements of the training data may be perfectly recoverable from the latent space, but most won't be. It's not very surprising as a result that you can get it to reproduce copyrighted material word for word.

I think you are confused, how does any of that make what I said a lie?

I can do that too. It doesn't mean I directly copied it from the source material. I can draw a crude picture of Mickey Mouse without having a reference in front of me. What's the difference there?