Theoretically what is supposed to happen is that the production class is supposed to fill the inventories of local vendors, then spread throughout the caravan network after they reach a certain capacity. I have only seen this function on nearby settlements (for instance abernathy and sanctuary, or finch and slog). It would require a lot of testing to see if it works out, but I think a key problem is if you have vendor restock set to "ON" then the vendor inventories are reset. This is problematic for production because it erases all those items you spent resources on producing in your production plots every few game days and it also prevents the vendor inventory from reaching whatever the point it is where it is supposed to start spreading through the network. For these reasons, production plots don't really deliver everything they would ideally as I don't think the functionality works the way it's supposed to.
I will say, to
@yaugie 's point above - there are many production plots that are useful though in terms of what they produce or that provide a lot of interesting in-game "flavor" (i.e. New Bugle to read tales of your own exploits, Magazine Publisher to get new perks - some of which are pretty good). In those cases I will have a settlement specialize. For instance, put a bar in a settlement with a brewery and a distillery so it benefits from both. Put a weapons store in a settlement with an ammo press. But the Wasteland Tailor in a settlement with a clothing store, etc. etc. This gives each settlement some extra purpose if you are looking to buy certain things. You don't need to have every kind of store at every settlement.
However, as you indicated above, you can still have a settlement with every store and it will still generate caps whether you have production classes to generate inventory or not.