Very good question. Birds need to keep their feathers in very good condition for flight and to keep them waterproof. Ostrich chicks, however, can die if they are soaked by rain and their downy feathers are wet. Dr. Alan Feduccia, a biologist who opposes the dinosaurian origin of birds, pointed that out. He then suggests that a dinosaur with downy feathers would be maladaptive for the same reason. It makes no sense for a dinosaur to have insulation feathers, if rain will soak these feathers and give the dinosaur a life threatening chill. Of course, logic is wasted on those who are dogmatic, and the believers of the dinosaurian origin of birds are indeed dogmatic.
The so-called feathered dinosaurs fall into two groups:
1. Those with real feathers and they are almost certainly birds, not dinosaurs. Included in this group are such fossils as Caudipteryx, Protarchaeopteryx, Sinornithosaurus, Microraptor, and Anchiornis. Feduccia pointed out that Caudipteryx and Protarchaeopteryx are actually flightless birds and that they do not have a single uniquely dinosaurian character. A few days ago, a group of scientists published a paper in Science and showed that Yarnornis, Confuciusornis and Sapeornis are birds that have 4 wings. Before that publication, only Microraptor, Sinornithosaurus and Archaeopteryx are known to have 4 wings (2 forelimb wings and 2 hindlimb wings). All these fossils (some classified as dinosaurs and some as birds) with feathers and 4 wings strongly suggest that they are closely related. That means Microraptor and Sinornithosaurus are either the closest relatives of birds or they are actually birds, even though some paleontologists classify them as dinosaurs. Feduccia, for example, considers Microraptor a bird.
2. Dinosaurs with dinofuzz. The first dinosaur identified with dinofuzz was Sinosauropteryx. Excited by the find, a group of scientists (dubbed the dream team) went to China to examine these fibers found on the outline of Sinosauropteryx to see if they were really feathers. When the dream team came home, not a single scientist belonging to this team would state publicly that these fibers were feathers. Not one of them will stake their professional reputation to make that claim. The exception was feather expert Alan Brush, who was about to retire. He claimed that these were "protofeathers." Protofeathers can also mean scales, since feathers evolved from scales. Therefore his statement can be viewed as equivocal at best. Nevertheless, these "protofeathers", which some scientists, including Feduccia, have said were most likely collagen fibers (the same stuff that aging humans inject into their skin to keep them looking young), began to "evolve." Before we know it, what was called protofeathers are now called "feathers" and all sorts of dinosaurs (including T. rex) are now being painted or reconstructed with feathers. Not surprisingly, dinofuzz (i,.e. "feathers") has since been identified in a carnosaur and even an ornithischian dinosaur, even though these animals have next to nothing to do with the theropod dinosaurs that are supposedly the ancestor of birds. If feathers are so widespread among dinosaurs, then it becomes a shared ancestral character between the birdlike theropods and birds, instead of a shared derived character. That means using this character alone, birds cannot be considered more closely related to Velociraptor than it is to an ornithischian dinosaur.
So, in conclusion. There are bird fossils, with feathers, that have been misidentified as dinosaurs. At the same time, some dinosaurs found with collagen fibers preserved are misidentified as "feathered." If that is the case, the dinosaurs really did not need anything to preen their collagen fibers, and the birds of course can take care of their feathers just fine. Your problem is therefore solved, because there is no dinosaur which had to preen its feathers, not anymore than any mouse needs to preen its feathers. Dinosaur feathers are therefore as real as mouse feathers. As for preening feathers with a jaw full of teeth, Archaeopteryx and other primitive birds had teeth, and they have feathers that have interlocking barbules, like modern feathers, but they apparently have no problem preening their feathers. Therefore, toothless beaks are not required to keep feathers in good condition.