Almost nobody programs in functional style in Common Lisp, even though it’s possible. In the vast majority of CL programs there’s tons of global state and yet one learns quickly how to architect and structure the program in order to make image based development and runtime code modification a breeze.
So no, the magic sauce is not “functional”, it’s that CL stemming from the Lisp Machine paradigm is designed for that sort of experience whereas something like Python very much isn’t. And it shows (there’s no image based development in Python that doesn’t end up creating more problems than it solves but image based development is not only the norm in CL but also tremendously empowering).
This is where the functional programming model really shines - there typically isn't a bunch of hidden state floating around to break hot-reloading