@OutByEaster?
As far as I'm aware, Chicano is a cultural identity name for Mexican-Americans in Southern California Arizona and New Mexico that gained widespread prominence during the civil and labor rights movements of the 60s and 70s. It was a cultural pride thing.
Is it offensive now, I don't think so, but these things can evolve quickly and I live closer to England than California so I might be out of touch.
LatinX is a new label embraced by some of the younger generations but I've seen it rejected by Gen X and older.
As far as the term being relevant to this story, it is because this is the first time there has been (Chicano) city councilors, including the president Leury Martinez, so it was considered a great opportunity to advance the interests of a historically underserved demographic in LA.
Also, one of the uglier things on the tape was their insulting the features of Oaxacans, who are very indigenous looking, typically shorter and darker as opposed to the more mestizo and Spanish features of most Chicano/Mex-Americans.
They also insulted a gay white councilors adopted black son who they called a monkey, among other racist things.