Griffins are crazy fierce.
There are lots of animals which would be useful domestic animals but which were not domesticated. Zebras, for example - they are built like horses, so why not? This synopsis of Jared Diamond's Guns Germs and Steel lists the qualities an animal must have for it to be successfully domesticated.
Not picky eaters. Ideally not competing with the humans for the same sorts of food.
Reach maturity quickly. Although elephants would seem to violate that rule.
Breeds in captivity. Again the elephants break that rule.
Docile by nature. This is the one that applies to zebras, Cape buffalo, American bison etc. They are bad of ass and tend to get mean.
Numbers 1 and 4 could apply to griffins. Maybe 2 also; see below. For interest in your story, you could have the fierceness kick in at puberty. You see lots of images of people cuddling and playing with baby bears, baby tigers etc. Less so when they grow up, because one grouchy day for the tiger means no more cuddling forever. Dingos are like this somehow - even though they look just like dogs and are sweet as puppies, they are untamable as adults.
If griffins mature slowly, your nobles could have baby griffins as house pets. The pups are not abused because they are prestige pets and then are put down when they grow out of their cuteness. This also is another path to domestication - via neoteny. This is most famously demonstrated in the case of domesticated red foxes. At a Russian fur farm, foxes which were less fierce were kept as breeders and the fierce ones used for fur. Over the generations the foxes got more and more puppy like and less and less fierce. They are bred to be tame animals. There are popular videos of these charming tame foxes.
If your story can use it, you could have the same process with your griffins - the tame ones are like overgrown pups and physically different (and smarter) than their wild cousins. The few rare adults which are tame are too valuable to be abused - they are intelligent and formidable companions, and also used in the hope of breeding more like them.