Sure, he wouldn't, but NBC would seemingly take whatever opportunity it could to crow about the fact that they have Josh Elliott and ABC doesn't. I don't watch Today very much anymore, so I don't know for sure how often Elliott's on the show, but it seemed to me that he's not on the show all that often for someone who NBC would seemingly want to inundate Today viewers with if he is indeed Lauer's annointed successor. That's not to say Elliott's not in line for it; it just doesn't seem like it's a sure thing right now.
According to Wiki, Vieira announced in April 2006 that she was taking over for Couric, she officially started in September 2006, Today announced in January 2007 that they were creating a 4th hour, and the 4th hour officially began September 2007. So, your initial timeline recollection is pretty close. I think it was more Curry than Morales who they were trying to appease- Morales was pretty new to Today, having just come over to Today fulltime in 2006 from MSNBC.
You may be right about the co-host thing; I may just be recalling what you say, that they had constantly rotating co-hosts because of frequent absences by Curry or Morales. It was almost always at least 3 co-hosts, I think, and I recall that Amy Robach and Peter Alexander were on a lot, so they might merely have been filling in. It was definitely a literal roundtable though, at least at the beginning, as this clip from the debut of the 4th hour shows.