Hi Ismene, welcome to Pharaoh Heaven.
The cart pusher of a raw material producer (such as a reed gatherer or clay pit) will deliver their good to a workshop that need it (such as a papyrus maker or potter or brickworks), and will deliver them to a storage yard otherwise.
The cart pusher of a storage yard will go get a good if the good is set to "get", the storage yard doesn't yet have the specified amount, and the good is availabe in another storage yard that is not set to "get". Otherwise, if the storage yard has one or more deliverable goods then it will attempt to deliver one.
There are several potential problems with deliveries. First, check the obvious possibilities--a missing road connection (or a building touching multiple unconnected roads), a building not fully staffed, and a good set to stockpiling--which are easy to overlook.
A storage yard can support several workshops if the delivery distance is short. If a lot of workshops that use the same raw material are far apart, then a storage yard (or producer) may end up delivering to a distant workshop. Therefore, it is usually best to have all connected users of a raw material close together.
In general, it's better for a storage yard to contain at most 1 good that is set to "get" or might be delivered. It's sometimes possible for a storage yard's cart pusher to start delivering a good but to not succeed (because another cart pusher delivered before him). If there isn't anywhere else for the cart pusher to deliver, then he'll stand around waiting for a delivery opportunity. Meanwhile, if the storage yard also contains another good that should be delivered, that other good might not be delivered for a long time.
Disconnecting the dock area from the rest of the city is often useful. This prevents dockers from going very far and therefore keeps the docks fairly efficient. If workshops using an imported raw material don't fit into the dock area very well, they can be put a moderate distance away (not connected to the dock) and supplied by 1 or more storage yards that get (typically, "get 1/4") of the raw material. Storage yards in the dock area can get river exports from the rest of the city. My Rostja is a good example of a city with an active disconnected dock area. (Nowadays, just for fun, I typically connect all of the roads in a city, which often makes the design considerably more difficult.)
With a good design, storage yards generally don't require any vigilance. For example, a storage yard that should have room to store beer should not be allowed to fill with pottery.
When you're both selling and using a good, it's sometimes hard to set "Export over" orders" and storage yard "Special orders" to make sure that housing will always have enough of the good. (It usually can be done, but can be quite complicated.) If there is plenty of money to be made in other ways, then you might consider not selling a good that houses use.
The "Brugle storage yard" is sometimes helpful, but I've used it in only a few cities. Many well-functioning cities were built before (and after) it was described. I suggest that you learn to use storage yards in the usual ways first. And by the way, if there are weavers then flax would not be a good "filler" good in a "Brugle storage yard".
I'll use my Abedju as an example. The 12 breweries are all close to 2 barley storage yards. The 12 weavers are all close to 2 flax storage yards. There are too many clay-using workshops (22 brickworks and 4 potters, although a few of them sometimes lose labor access (I didn't design as well back then)) to put then close together, so there are 4 unconnected groups of clay-using workshops: 1 connected to the docks and supplied clay by a storage yard close to the dock, and 3 others supplied clay by storage yards set to "get 1/4" clay. (Each set of brickworks also has a storage yard delivering straw, but since a brickworks uses 1/4 as much straw as clay, straw deliveries are easier to handle.) The papyrus makers aren't as concentrated, but none are a long distance from a reed gatherer, and no storage yard contains reeds. None of the workshops that contain a raw material contain anything else that might be delivered, but some of them contain a good that isn't delivered, such as bricks or fish.
My Abedju doesn't really have a disconnected dock area, but there are several unconnected parts of the city. The docks (I used multiple docks back then) are connected to the breweries, weavers, and papyrus makers. There is only 1 storage yard containing beer and only 1 containing linen (both near the dock), and only 1 containing papyrus. Only 1 housing block uses beer or linen or papyrus (and is, obviously, connected to the dock). There are 2 other housing blocks that use pottery (not connected to the dock or each other), so there are 3 pottery storage yards (1 "accepting" and 2 "getting").
You might find it helpful to look at a few example cities from our Downloads, perhaps cities that you've completed (to see how other players handled the same problems).