“Moves to advisory role” is just corporate speak for “retired/fired”. I can’t think of a single example of an executive who “moved to an advisory role” and demonstrated even an iota of influence after that point.

Right, that's corporate speak. Welcome to gov speak:

> A Special Government Employee (SGE) can perform temporary federal duties for up to 130 days within any 365-consecutive-day period

https://www.flra.gov/Ethics_Rules_for_SGE

As an SGE, the person has legal influence, but ethics rules are relaxed. As an "advisor," there are almost zero ethics rules, and their influence is not legal, but wink wink. Many of the most influential people in the current gov are "advisors."