There is clearly value in a trained AI that an untrained model lacks, otherwise you could sell them as a product or service for the same price. That training has value, and price difference between a trained and untrained model is that value.

Because training has a value, the training material has value as well. You can't commercially extract value from someone's product to make your own product and sell it, unless you buy their product wholesale or through a license.

And it they argue that paying would be financially prohibitive to training, they admit that the training has financial value. It'd be cheap if the training material wasn't valuable.

I see two likely paths here for the future, presuming the court rules in favor of the NYT. The first is that AI companies work out a deal with publishers and media companies to use their work while not breaking the bank. The second is that AI companies don't change the training process, but they change their financial model -- if the AI is free to the public, they aren't making money off of anyone's work. They'd have to charge for ads or something.