We know they do that. It's stupid that the woman across the road complained, but I'd say the hostile neighbor is the problem in this scenario -- not the HOA. The guy delivering the notice has no choice in the matter.
People in HOAs who deliver the notices have every choice in the matter. They are the ones directly upholding the covenant/HOA bylaws by participating in the matter. It's a living entity, not static. Those delivering notices without also getting involved in the discussion are just as much in the wrong as the original complainer and the HOA itself.
I live in a small HOA, technically a condo association, and I have few qualms with it. I mean I have some, certainly, but they are largely small issues. In return for my dues, they upkeep the entire property (outside the walls), and they also keep the external walls of the buildings including shared walls. Meaning any leaks and necessary repairs on any of the 4 walls that make up the individual unit, as well as foundation issues and even the main drainage I believe.
Barbecuing of any kind has never been an issue that I'm aware of. You can't have open fire pits on your patio, but that's a combo of state law (no open pits on residential property, likely based on proximity clause, so those with large enough property can have one I think) and the fact that it would definitely be a proximity issue anyway in my case.