I don't think his saying they put him through the ringer is a bad thing. He probably knows more than anyone else that he was just a token interview. Vieira being part of Arsenal's history does mean something. Sure, Arteta played for them, but he doesn't have the same respect or established credibility with supporters. If you're going to go for someone who is inexperienced (and they both are) then you're much better going with someone who might be able to get a bit more rope from the supporters before they're calling for their removal. There also is something to be said about the passion and knowing what the club means. Arteta knows a bit, and I think he'll treat it with some respect, but he doesn't have the same credibility.
If Klopp was to leave, I wouldn't want Steven Gerrard to be the manager of LFC. But at the same time, if it was Steven Gerrard or someone else who was inexperienced and basically had no management history, I would take Gerrard in a heartbeat. He knows what the club stands for, and while he might not be the best guy for the job, he also would do anything for this club. Being a legend, and knowing how close he's tied to it, I would at least be willing to give him a bit more of a chance.
And note, while I don't want to turn it into that debate, I do know there is still some stigma against some players ending up as managers. I don't think them not hiring him isn't completely related, but I do think it's something that the league needs to push for as they go forward. Obviously, the best manager option should get the job. But I do think there's a bit of a question on the fact that there's only Hughton in the EPL and barely any minority managers in the top 4 leagues in England and much of Europe in general. I do think there is some growth there that needs to occur at some point.