It is really annoying to claim the The Gadsden flag is some kind of hate symbol.
The "why" of the inclusion of the Gadsen flag in those types of releases is now in the public domain as it was at least partially discussed in the public media after another one of those alleged "police intelligence" reports was leaked. I can comment on it without violating any constraints.
It happened maybe two or three years ago. If I recall, it concerned some person or small group in PA (I think). Anyway, the Gadsen flag was used by the person or group involved. One of the Federal alphabet agencies wrote a release. Since the flag was used by the folks arrested or investigated, the inference was made that it was a symbol of such groups. As an aside, I seem to recall that "ΜΟΛΩΝ ΛΑΒΕ” was also mentioned.
It’s not the first time a symbol or motto has been “hijacked” by an extremist group and then that symbol or motto takes on a whole other meaning. Perhaps a more innocent example is the rainbow symbol. In the 60s, that was a symbol displayed by folks who wanted world peace. It was rather common in CA. Then the militant gay movement took it as a symbol of gay pride in the late 70s. They used different colors, but today, any rainbow decal is pretty much inferred to be a gay pride symbol.