This quest should unlock dragon riding, it’s fairly early on into the Waking Shores on the Dragon Isles.
You should get an extra button in the middle top of your mounts tab, this includes a toggle to swap flight stance and also gives you access to your flying skills.
If you still need to gather all the flying glyphs you can just fly around the dragon isles to pick them all up. You don’t need to quest to do this. You can just fly around to grab them all.
It is also worth noting they fixed this bug:-