This is the crux of the matter. If the HOA has a rule against flags, fine. (I wouldn't want to live under the thumb of a homeowners' association, but I concede that they are entitled to set their own rules.) If the HOA wants to allow only certain flags, as defined above, fine. But if the HOA is not enforcing its own rules, allows all sorts of flags to fly, and then steps up to sanction one viewpoint, the HOA is wrong.
It would be interesting to know if any BLM flags are flying there. Or any other flags expressing a fairly direct political point of view. If no one has crossed into political territory yet, the HOA might be able to salvage its rule. Around here, I see mostly American flags, college flags, Marine Corps flags, and Redskins and Nationals banners. There are, however, a few BLM flags and rainbow flags here and there. If that's comparable to the spectrum in this case, the HOA doesn't have a leg to stand on because the political barrier has already been crossed.
I don't think there's any such thing as BLM flags. Their territory is marked by burned out cars and buildings.
Mark