Yes, that's the goal that was used, but they are quite simplistic and crude methods, not some specifically designed function, with carefully fine tuned parameters or something. So, if a basic function like top_k can improve model utility, it's not impossible to imagine that watermarking could also happen to do so, or at least not have a significant negative effect. So whether the effect is deleterious or not is an empirical question, not something we can assume ahead of time.

> You're basically saying that if we train LLMs on watermarked text they'll be really good at producing text that looks watermarked

No, you're misunderstanding how the training works. If we train the model's output so that it minimizes the error function after the watermark is applied on it, the model will learn how to produce the best output it can given the watermark. It will produce better text that happens to be watermarked, not "more watermarked text". Same as if you train the model on minimizing `top_k_error(input) = |top_k_sampling(model_output(input)) - desired_output(input)|`, the model will learn to produce better output under top_k sampling, not learn to produce output that's "looks more top_k".