Just came across this bug, took a few dozen attempts to ride the gryphon - ended up working after a while whilst mounted.
Still bugged in July 2021. Here’s how I made it work:
Mount up. Stand INSIDE of the second gryphon to the right (stage left), facing same direction as the gryphon. Basically line up your mount with the gryphon. Click the gryphon. If he pops into the air and starts hovering, click him again. Hopefully this works for you, because it doesn’t seem Blizzard has any intention of fixing it.
that worked for me too, mounted then clicked the gryphon.