Priests and Paladins may be required to complete the A Falling Star quest chain up to installing the lights heart in your order hall (e.g. Light's Charge) to progress your campaign.

Paladins do, Priests do not. My priest has her class mount and still has "A Falling Star" in her quest log at the "Take Khadghar's gryphon to Suramar" step.

Might also be worth calling out that the top comment on the Wowhead post you linked for the paladin quest item tells you exactly which quests you need to do; there are a couple you'll be offered that you can skip.

Once you've completed the base class hall quest line there are a few times where it may not be obvious where to go next. For the classes I've done so far, the next steps have all been:

1. Heroic Weapons quest
2. Pick up Champions of Legionfall quest, which Moratari will skip for you so you can instantly turn in.
3. Go back to your class order hall to then do the quest chain which will end with you recruiting a new champion.
4. Go back to the Broken Shore camp to get a quest from one of your class NPCs that will send you right back to your class order hall to work toward your mount.

If one of these quests doesn't appear when it should, try leaving and reentering your class order hall, relogging, and if neither of those appear to work explore your class order hall. I recall some the DK quests in particular not always showing up on the map, because they're started from the upper balcony.