It's one of the first questions on my list for the homes I'm currently looking at.
HoA's are almost universally bad news. At least for me, as a drummer who enjoys jamming with other musicians, grilling enthusiast, and fan of backyard firepit gatherings.
But that said, I can't watch the video right now, all I can do is read the article, and the article says he's using commercial equipment. Which, to me, says he's got an industrial-size BBQ going on in his back yard which could actually produce more-than-slight problems for neighbors in close proximity.
I have a regular largish charcoal grill that I use once or twice a week and while my neighbors are really cool with it, if they weren't I don't know if I could rightly defend myself to the property manager because that sucker does sort of make a lot of smoke for a bit when I light it up that could seriously interefer with others on their back porches.
I can believe that if he's using commercial-grade equipment, and the dominant wind in the area constantly pushes that smoke on one neighbor, there's a problem that needs addressed.
Please keep in mind, I'm only defending here because even this article (from sensationalist site) takes time to note that he's using commercial-grade equipment and that it references the systems restaurants use to control such emissions...which are going to be out of the price range of your average home griller.
Edit: OH. And anything else aside, this is not a "nanny state" issue. This is a dispute between neighbors where the government official(s) involved were required to follow-up on a complaint and determine the appropriate course of action. Maybe the official(s) in question subjectively got the call wrong -- and if anyone is getting cited, I agree that they did -- but this isn't the government (on any level) cracking down on something as a whole.
And while even if they did get this call wrong, I've heard stories of (and even witnessed) some really awful neighbor issues where the local government wouldn't step in. Which is a far, far worse situation, let me tell you.