This is more about the dragon tutorial quests than anything else (it's been a while since I've tried the later quests, but I will comment on how they were under the presumption they're the same).
Now I don't know exactly what the capabilities of the game are, but I do know it has the capability to post private text to the player and update quest steps when you reach a certain location, and I think this could be used to reduce the time you spend running back and forth. This is particularly bad during the dragon tutorial in New Trismus; the dragon crafting trainer is particularly bad for sending you back over and over again to the same few spots. I think if rather than having to go back and speak to the trainer over and over after each material acquisition, you'd be routinely guided to gather all your items at once, it would help reduce the grind.
Note this isn't as much of a problem later as during later craft quests the game no longer logs gathering resources, but just tells you to 'make an item', so it's primarily the earlier quests. That said, if there was a dragon teacher closer to the materials or if Kerian could move to the area while you're on these quests, that would be a little more interactive, but I'm not sure what the system is capable of doing with NPCs.
My second suggestion is to allow players to take multiple quests from an NPC at once. If I remember correctly while playing my first dragon, you were only allowed to take one quest from each NPC, or have one training quest (Gold rage quests, etc) active at a time. I never really understood this and always put it down to padding out the time taken, but I think it would be a nice quality of life improvement when completing quests if you didn't have to backtrack for each one.
I understand that HZ in particular might have issues with this either for technical reasons or that it will make completing quests faster due to saving on travel time, but just tossing it out there.