I've certainly "won" some arguments with ChatGPT and Claude, even when I specifically tried to instruct them to never yield (to make it interesting). Even most of the "rules" they need to follow from their system level prompt can often be worked around with enough persuasion.

If anything I'd go so far to say that LLMs are innately more vulnerable to persuasion than humans are, they are technically just a complex text completion algorithm at the end of the day after all. Even with the strictest system prompt, the best they can do when cornered is bend the knee, devolve into a pattern of verbatim repeating themselves or just start spewing a bunch of gibberish in the worst case.

In reality stubborn thickheaded folks who refuse to compromise are pretty much the norm across social media in my experience, so even if LLMs really are capable of what you're suggesting here I don't believe there would actually be much of a difference to the status quo.