The fractal nature of lithium ion battery electrodes interfacial area is currently ignored, and I understand why: what can we do with, for example, the fractal dimension of the surface area (what for, from a very practical point of view?)

Still, that's something discussed in the literature, a little, myself (doi:10.1149/1945-7111/ab913b) but also in the fuel cell community (see Bertei, A. et al. The fractal nature of the three-phase boundary: A heuristic approach to the degradation of nanostructured solid oxide fuel cell anodes. Nano Energy 38, 526–536 (2017)).
Currently I think it's important people in the field understand that surface area estimation from 3D imaging (or length for the triple phase boundary) is a fractal property and therefore the value calculated from imaging is to be considered with caution (i.e., do not use as it in a model... you are probably off).

I do not have enough background in fractal analysis (although from what I have read calculating the fractal dimension is not that hard), and then I am struggling finding a way to make a practical use of a fractal analysis. By this I mean, what can we say about two electrodes with different fractal dimension, except that they have different surface roughness? Can we correlate the fractal dimension with another property that speak more to the battery community? Maybe we could relate it to surface degradation mechanism such as the SEI growth? I do not know.
I would greatly appreciate if someone with a fractal background could help.