> How have you or anyone been influencing or analyzing the randomness behind the sampling process to create better writing?

You're mischaracterising or misunderstanding my point, or I mangled it.

I mean it is possible to analyse, control, monitor, study the impact of changing temperature on the writing, yes?

The point about watermarking is that this relationship — change the temperature, see the effect — is now being adjusted by an unstated, secret process you explicitly can't control.

(I gather Anthropic have recently taken away this setting anyway; that was news to me.)