Yes it is. And you don't roll that if you're trying to persuade someone. You roll your persuasion vs the DC set by the GM.
I didn't say it is. I said it is the rule that the GM sets the DC to represent the difficulty of the task. How do you judge the difficulty? You need to take into account the factors I mentioned earlier and more. So not every argument will have the same DC. The players might keep the conversation going to get a better read of the NPC. Some insight rolls might be involved (keyed to different ability than persuasion, so the best persuader might not be the one who succeeds.) Then the player who thinks they have an argument they believe would convince the NPC makes their pitch. So they're the one who is rolling.
Also a conversation might involve several lies and arguments, so it is perfectly possible that several rolls are made.
Like I get the idea of having a face character. If you already have a clear read of the situations and a straightforward request with a clear pitch it makes sense for them to do it. But more complicated social situations just don't work that way.
Furthermore, this is roleplaying game, not a competitive game. The point is to do what makes sense for your character. So just have conversation in-character like real people. It is not about winning. Hell, I loved playing an irreverent barbarian who often said things that annoyed NPCs. Might not have been optimal, but it was fun.
And yes, all this certainly is me extrapolating beyond what the books offer advice for. Because, frankly, the books are rubbish at that. I wish they were better, but they aren't. Good thing that a big part of newer gamers are familiar with Critical Role and other such shows so that they can get an idea of how it is done.