There are several approaches to hanging loads on drywall/plasterboard.
Light Duty - The most common anchors for used for light loads are expanding tubes that grip the sides of the hole they are in and flare lightly behind the hole
Others like yours, have a wide thread to grab more surface in the drywall itself. Some of the these threaded anchors have a flaring end, some do not.
The limitation of all of these types is they depend on a small area of drywall to stay solid to hold them. They are good for light loads and when the main force on them is shear force. In a wall application, that means most of the force is pullin down, along the wall (as if you are trying to cut the screw in half right at the wall).
If you put a heavy load on these anchors, there is significant outward force, that is, pulling away from the wall. In a coat rack some of the load is shear (downward) and some is outward. This puts stress on the sides of the hole in the drywall and it crumbles. This is even worse when the load is dynamic (moving) rather than static (non-moving). A hanging coat is static. A coat being pulled form its hook is dynamic. And almost all of that load is outward.
The threaded anchors distribute the load better, but not enough for any significant load. Sometimes this is compensated for by using many anchors. An improvement, but usually not a good solution.
Plastic (or metal equivalent) anchors are almost never suitable for ceiling mounts, where all the load is away from the ceiling surface.
Medium Duty - Anchors for medium duty pull against a larger area behind the drywall to distribute the load.
Expanding anchors go into the hole as a tube and are then expanded behind the hole using a screw, bolt or tool.
There are also toggle type anchors where the expanging section either rotates or springs out to cover a greater area.
The advantages of these anchors is that they spread the load far beyond the hole in the drywal itself. While the hole may be prone to crumbling, these anchors rest on an area of solid drywall, from about 1 to 2 inches around the hole. This gives the anchor much more strength to deal with outward loads of medium duty, as well as shear strength. However heavy outward loads, especially dynamic loads, are pulling on on a few inches of thin plaster and paper. Again, adding many anchors helps spread out the load.
Some of these anchors allow removal and reattachment but some do not (for example, spring toggles fall off in the wall when the bolt is removed).
Heavy Duty - On a plasterboard wall, the heavy duty mounting of choice is directly into framing members.
The most common approach is to find studs, the upright framing members that hold up the wall surface. These are either wood or steel. Screws are driven throught the item to be mounted, through the drywall and into the middle of the stud. An alternative is to screw into cross bracing framing. These are wooden or steel sections running horizontally to brace a wall or serve as a firebreak. These are marginally less strong than upright studs, but for practical purposes can hold all but the heaviest loads.
Studs and cross bracing can be found using a stud finder.
Screws should be long enough to reach at least 1 inch into a wodden stud, longer if the load is especially heavy. In steel studs, finer thread screws should be used (sheet metal screws) and length is less of an issue.
In wooden framing, an alternative to regular screws are hanger bolts that can be threaded into the studs and leave a bolt on the outside of the drywall for hanging the object.
The advantage of hanger bolts is the ability to easily remove and reattach the load using a nut or wingnut.
When there is no framing member in the exact spot needed, it is sometimes worth the trouble to add one. A section of drywall can be removed from the area, a cross brace inserted exactly where you want the mounting and extending horiziontally to the two supporting studs on either side. A drywall patch can be then put back to cover the cross brace (using the drywall section removed or a new piece).
Best Answer
My default fix would be just to put in some type of anchor. Given the damage and the fact this is on a ceiling, I'd be inclined to use a screw-in type anchor, because they hold on to a larger area and would deal better with some damage.
If that couldn't work, the next thing I would look at would be getting a new surface to screw into.
Normally the method is to use a backing board, which is basically just a small piece of wood -- perhaps 3/4" x 2" x 4" -- to screw into on each side. Unless the vent can be moved easily, this will be difficult.
In this case I'd actually construct some "wings" using some sheet metal to wrap around and over top of the drywall, which the vent would then screw into. They'd anchor into the vent itself.