Basically any modern project management style focused on people and needs instead of reams of documentation.
I've unfortunately never worked at a place that does this right, but I had the strong impression some really good companies do it right - I was blown away by Guidewire back when I was in the insurance business, for example, they really seemed to do project management right ( see https://www.guidewire.com/blog/best-practices , and stuff like this - https://www.guidewire.com/blog/best-practices/implementation... , Scott Hatland really impressed me as a guy that lived this kind of person-first approach ).
Since it's more of a people and organization thing than a formal "project management style" thing, I don't know if there's one "style" that does this above all - Agile, Scrum, Kanban, can all be twisted by people who don't understand the core concepts.
Unfortunately, in practice, Agile almost always devolves into a cargo cult: perform the ritual of the Standup at the appointed time, etc. and all will be well. Any actual personality management is papered over with appeals to authority.
I think teams that succeed do so despite using Agile (or any other management fad). A cohesive team that communicates well can succeed using Agile, sure, but the process is incidental; they would succeed using most any methodology. Survivorship bias leads people to believe that Agile (or whatever methodology-of-the-week is in play) is causative, when it is actually epiphenomenal.
Agile is hardly ever done properly. It doesn't fix the problem with having middle managers who form master/slave relationships, destroy team morale with bad decisions and credit pinching, or deal with the issue of having engineers who purposefully write bad software to "prove new technologies are not as good as old, tried and tested ones".
