In general, if you perform copyrighted works you are doing copyright infringement. There are certain exceptions (personal use, education, very small fragments with proper attribution, maybe a few others) but whether you are reading it aloud from a book or performing it from memory makes no difference.
So, if you setup a service like ChatGPT but powered by humans responding real time to queries, and these humans would occasionally reproduce large chunks of NYT articles, they and the service itself would be liable for copyright infringement. Even if they were all reproducing these from memory.
Now, this is somewhat different from the discussion of whether training the model on the copyrighted data, even if it had effective protections from returning copies of it, constitutes copyright infringement in itself. I believe this is a somewhat novel legal question and I can think of no direct corollaries.
I certainly don't think we can just handwave and say "at some level, when a human reads a copyrighted work, they are doing the same thing", because we really don't know if that is true. Artifical neural networks certainly have no direct similarity with the neural networks in the brain as far as we can tell. And, even if they did, there is no reason to give a machine the same rights that a human has - certainly not until that machine can prove sentience.