Michael's power base in 3 is never credibly explained. On the one hand, Zasa commands what used to be the Corleone street operations in NYC (the "olive oil business"), as he told Connie and Vincent at his party.
TB, as I've posted elsewhere, in III I see Michael as analogous to Tony Accardo in Chicago. Tony bossed the Outfit for so many years, then turned it over to Giancana. But Tony never lost his influence and ended up deposing Giancana. I think that Vincent's "Supreme Cousrt" statement about Michael's influence and the efforts of Michael's enemies to rid themselves of it sums it up.
