Unpopular opinion: Draco's arm really was seriously injured in book 3, and the accident wasn't his fault. First of all, the seriousness of the wound.
Hagrid was wrestling Buckbeak back into his collar as he strained to get at Malfoy, who lay curled in the grass, blood blossoming over his robes. Hermione ran to hold open the gate as Hagrid lifted Malfoy easily. As they passed, Harry saw that there was a long, deep gash on Malfoy's arm; blood splattered the grass and Hagrid ran with him, up the slope toward the castle.
- Harry Potter and the Prisoner of Azkaban, chapter 6
That is not just a little scratch. And he was in the hospital wing overnight, which I seriously doubt Madame Pomfrey would allow if it wasn't genuinely necessary. Sure, Draco told Harry and Ron he was milking it for sympathy, but consider this. His right arm - his wand arm - was in a sling, meaning that he was pretty much defenseless both physically and magically. Would you want to admit that to your rivals, especially when at least one of them is known for escalating verbal conflicts to hexes and physical violence?
Now, as for the culpability. Yes, Hagrid said not to insult the hippogriffs, but he specifically mentioned nonverbal cues like bowing and making eye contact, which Draco did, and was in fact petting Buckbeak when the attack happened. He did not say that hippogriffs could understand English, and he also slapped Buckbeak on the rump to get him to fly, which I wouldn't exactly call respectful. So, it was Hagrid's fault, both for using creatures too advanced for a third-year class and for leaving out a crucial piece of information in his lecture. Thank you for listening to my rant.