McCullum succeeded where Taylor failed because he purged the "senior" (I use this term because they were but ****in how?) players after Cape Town, the same players who desperately wanted the coup lol.
One of the few areas where the stories match between autobiographies is those players were pure ****in' cancer. McCullum doesn't name as many names or dish it out as brutally as I've just said but it is pretty apparent Jacob Oram, Kyle Mills, James Franklin, Jeetan Patel and one or two more I've forgotten were, and I'm paraphrasing a McCullum quote "not on board with the team ethos Hess and I wanted moving forward". In the ODI series he brought in a lot of new blood including Neesham, Anderson and McClenaghan.
McCullum was also partly helped by those guys being close to retirement otherwise after a couple of years trying to lift the team to be more than it was like Taylor did, he would have been out on his arse like Taylor. The senior players didn't want McCullum captain because he was better per se (though his people skills with young jocks seem top class), they wanted him because they thought he was one of their boys and would enable them to continue taking the mick instead of the Taylor/Wright/Buchanan (and prior to them Vettori) group constantly demanding more and Taylor specifically wanting to bring in more young talent, which triggered the named Jacob Oram back of the bus wahhhhhh.
The other thing which made some improvement inevitable was under Taylor's reign, the new talent his group brought in was quite good. Boult, Williamson, Watling, Bracewell etc combined with Taylor, Vettori, McCullum, improving Southee and sober Ryder to make the core of a decent team.
McCullum might have accelerated it through his personality but his best decision was political and it was the correct one. Purge the people mentally holding the side back.