If it's about* proving that people are getting around the paywall with OpenAI, won't it be much easier to prove this with a live reproduction in the court?

* I am not too familiar with this matter and hence definitely am not rooting for one party or another. Asking this just out of technical curiosity.

No, because OpenAI can change their server at any time - and does, to patch over the cases where their copyright infringement is obvious.