I presume that the investigation into Weiner’s sexting got the emails through a subpoena and alerted the FBI of the connections to the Hillary investigation - at least as the public reason for re-opening the investigation.
I suspect that the white hats in the FBI knew from the content on the laptops that Huma was central to a lot of the criminal stuff (they may have seen some cc’s to weiner on some of the emails too) and gave Comey the “idea” to use Weiner’s connection as the reason to re-open the investigation, rather than spilling out the truth about Mills breaking the terms of her immunity deal and Comey obstructed justice in ordering the laptops destroyed (since hiding that truth was the motivation to get Comey to re-open the investigation in the first place...) Having Weiner involved would be sure to get at least some media to cover the story.
I think your synopsis is as good as any I’ve heard so far!