I just noticed the excerpt about the actual "consignu" wasn't included in the excerpts I posted above and that's one of the most important parts haha.

So the operating structure of the Chicago Family circa 1969 was:
- Boss Giancana fled and at this point may have been taken down.
- Underboss Battaglia was briefly acting boss but went away.
- Ricca and Accardo serving as acting bosses.
- The consiglio meets every two weeks, consisting of Ricca, Accardo, and the captains.
- Alderisio sometimes attends consiglio meetings, apparently so he can then update other Family members about "activities and plans".
- Alderisio is seemingly a soldier direct with the admin (as he had been under Giancana) who in addition to serving as a liaison to the consiglio, also represents the Family in business matters. (We know from Bomp's earlier visit to St. Louis that Alderisio represented the Family in relation to a Teamster's loan, which adds to what Bomp said here.)
Alderisio's position was very much like any soldier assigned direct to the admin, where in addition to serving as a high-level messenger and representative of the admin, he's also a force unto himself. We've seen this many times with guys in this position in other cities. Like Caramandi said when he got assigned direct to Scarfo in Philly, it's one of the most advantageous spots you can possibly be in to be direct with the boss.
- Also note that the description of the "consignu" as the "Commissione of the Family" is telling, as the Commission isn't the "administration" in the usual mafia sense, it's a panel / committee of select bosses. Similarly, the consiglio is not the administration -- it includes the administration but is a council of top leaders and senior figures of importance who vote and discuss major decisions as a panel.