When does baking/cooking in bulk have an advantage?

Mostly, it's just more efficient from a time perspective. You can do all the stuff that a restaurant kitchen does at home on a smaller scale, but it may seem like a lot of work for just a small portion.

For example, the prep time to make homemade broth, then put together ingredients and cook a homemade soup for 4 servings might take you most of an afternoon. Maybe by adding an extra hour for processing ingredients, you could easily make 100 servings of that same soup. The difference between making 1 loaf of bread and 20 loaves of bread is minor, assuming you have equipment to handle the larger quantities. You'll spend a few more minutes dividing and shaping, but otherwise most of the time is just waiting for the dough to rise and then bake.

So it's really about efficiency. If a recipe says "stir for 20 minutes while simmering," you'll probably have to do that for roughly 20 minutes whether you're making a cup or 3 gallons. And because all that prep work is tedious for just a few servings, many home cooks won't bother -- they'll substitute store-bought broth or bullion, they'll buy pre-mixed bread dough, etc. Those "shortcuts" are often one main difference that can make your home cooking not as good. Otherwise, there really aren't specific advantages for most dishes. In some cases, you may need to adapt techniques or recipes to handle larger vs. smaller quantities. But flavors can be effectively developed in most dishes whether you're making one portion or a hundred.

Fermentation is more active in larger batches, and at scale you can backslop to sustain operations instead of starting from scratch with a yeast packet (or hoping for the best from whatever state your tiny sourdough starter is in). Also feedings, a larger operation can more easily afford multiple feedings of sourdough per day, while someone who makes a loaf or two per week would likely stick to only one feeding.

A couple additional advantages in large batch I can think of, temperature control can be easier for processes calling for steady temperature or slow changes as adding an ingredient to a small batch will have less effect on the temperature of the batch. It is a disadvantage of course if you need to rapidly change temperature.

Larger batches tend to be more forgiving of measuring errors. Something calls for a level teaspoon and home in a two cup recipe and you use a rounded or short teaspoon it may be more noticeable than in a large batch calling for a level cup and you slightly round or short it. Large measures also tend to be less prone to error especially if you can use pre-measured amounts like a pound of butter rather than an ounce.

There are also some processes that just will never work as well in small batches, such as how does one prepare a single serving of a standing rib roast and duplicate the effect of roasting a full or half rib? In general I think you hit some of the types that can benefit, slow processes like soups, items like bread where a home recipe might call for a pinch of salt but a large batch can have a much more accurate measure and add in to the category things which need more of a temperature reserve and even application than you can provide at home.

Something like a broth. A high end kitchen would make a fresh broth daily. At home you probably would not. If you have and take the time for all the steps then I don't think a restaurant has any advantage.

There may be some suppliers that will only sell restaurants. You may not have access to some of the same goods.