Turnbull....That was "extremely" well spoken. And I think you nailed it as far as Lansky's overall influence and the power he wielded in the American underworld.
In his younger days, while he was coming up, he did "work," he ordered "work," as you said. But he was never the orchestrator or originator or those plots. He worked in tandem with Luciano and the others, and facilitated the work they ordered.
He worked with nearly all the different families and crews, yet, he himself was a top "associate" of the Luciano/Costello/Genovese Family. The most powerful! So he essentially had card blanche to operate, so long as he cut them in for a piece of the action. Which he always did. Etc., Etc. Etc.
The chart I designed, of Lansky's Syndicate, represents a good majority of those Jewish and Italian mob guys affiliated with him over the years, many of whom looked to Lansky for tutelage and guidance. Many dated to his early days on the streets of NYC. But the "Lansky Syndicate" (for lack of another name), was always a loose confederation of diversified, but like-minded men. Not a formally structured, hierarchical "family" per se.
But how you worded your post was great, and I think it gives good insight into his true underworld standing. Bravo!