With this issue, the devil lives in the details. The detail at issue here is the term "commercial" in description of this person's kitchen as opposed to the term "industrial."
The health laws in all 50 states deliberately define the term "commercial". Although the language used is different from state to state, "commercial" is generally defined as any operation that involves the preparation, production, storage, or service of food items to the public. This includes not-for-profit and religious organizations, even if food is part of a business where meal(s) are included as part of the services rendered. The exception, of course, would be if the food provided is ordered from a separate restaurant or catering service.
The health codes are specifically written in this general way to close the potential for any loopholes. For instance, if the equipment in the kitchen, and not how the kitchen is used, formed the basis for an inspection, what's to stop a restaurant or any other operation from using all residential equipment in order to sidestep the health department altogether? To take the matter one step further, many states allow for home kitchens to produce certain "low risk" foods for sale to the general public such as candy, cookies etc. In such cases, an exemption can be applied for that would essentially do away with the health inspection requirement and allow the kitchen to produce these limited food products commercially. Again, its not the equipment that defines the term "commercial". Its the use. In other states, a home kitchen is not required to be inspected if the gross sales of certain food items are less than a certain amount. If I'm not mistaken, South Carolina has such a law. But, don't quote me.
On the other hand, an "industrial" kitchen is simply a kitchen that has "restaurant" grade equipment. As far as the law is concerned, any residential home can have a kitchen comprised of all "restaurant grade" equipment, including vents, grease traps, coolers, freezers, dry storage, sinks etc. etc. not be required to be inspected. The home owner's insurance company may have a thing or two to say about it, but, as far as the health department is concerned, they don't care until that kitchen's use meets the definition of the term "commercial."
Now, if a person were to own such a kitchen, its entirely possible that the home owner's insurance company may require an inspection for liability purposes as part of the insurance policy. We all know the hazards of grease traps and industrial grade stoves and the importance of fire suppression systems and so do insurance companies. I am sure those risks will be amply reflected in the insurance premium. But, as long as that kitchen's use does not meet the definition of "commercial" under that state's health code, the law would not mandate an inspection.
Like I said, its very likely there are key details missing from the OP's account of the situation. Having said all this, its pretty much 100% guaranteed that if the HD is knocking on this guy's door telling him that his kitchen is due for an inspection, that kitchen was or is being used as part of some sort of commercial operation. Otherwise, how would the HD even know the kitchen has industrial grade equipment? The only other plausible explanation is someone reported him to the health department for one reason or another.