My understanding is that "stable" is simply that. Not right, not approved, but... stable. A non-vandalised version of the current consensus on the article - you'll see the revisions from before someone vandalised it, or before a violent edit war kicked off, as the stable version, but it doesn't mean they need to have been individually factchecked.
That's exactly my understanding too. It would be good if using the same system an article could be marked as "fact checked". We could come up with a system for experts in the field to read the article and agree that it is factually correct. I think we should implement such a system for featured articles (existing ones, not candidates - this isn't intended as another hoop to jump through to become a FA) at the very least. It doesn't need to wait for the software to know how to do it, a system using templates should be fine.