My character is level 62, and I have completed the relevant quest chain to unlock dragon riding beyond the point where I would have unlocked the ability.
I was never given the option to select a dragon riding mount (I already had a Windborne Velocidrake, unlocked on a Pandaria: Remix alt), and I have absolutely NO idea how to proceed at this stage.
Does anyone have any ideas as to why my Dragon Riding skill is faded out, and my dragon riding mount is acting like a normal mount?
I suspect that I was supposed to pick a dragon riding mount on my level 62; but, as stated, at no point did any quest give me that option.
I canāt continue to play the game unless this issue is resolved 
You should be introduced to dragonriding in the first zone of the Dragon Isles.
This links the quest chain required to be introduced to dragonriding as well as the specific quest.
Thank you for replying. However, I have completed this quest; I am up to āāDragon Glyphs and Youāā - which I cannot complete, due to my character being grounded.
I was never offered the choice of a Renewed Proto-Drake. The ONLY dragon riding mount I possess is a Windborne Velocidrake, chosed by an alt in Padaria: Remix.
I am extremely frustrated by this situation. I can see the Dragon Isles Summary next to my mini-map, but with regards skills and unlocks, it says āDragonriding skills have not been unlocked yetā. Under my general skills (skillbook button), I do have Dragonriding (Passive ability).
This seems like a major bug to me, yet contacting Blizzard support via their ticket system is far harder than it used to be
Oh, and also, I canāt use the customization point either; I had to go to Dalaran and change my hairstyle to complete one quest, as a workaround for the questline wanting me to customize my (non-functioning) Dragonriding mount.
Iām tempted to quit the game again, as Iāve only just come back. To face this issue as a solo player is very offputting.
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I would suggest at this point you should make a ticket. It may be that thereās something easily fixable thatās causing it or it may be more problematical.
If you get an automatic response and youāve already tried everything in it, then reopen the ticket.
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I believe my character is bugged. However, theyāve made it deliberately difficult to create a ticket - so as an actual PERSON will answer it, that is. As a result, Iām left feeling very upset by both the issue and the protracted process.
Thank you for your supportive reply. In the meantime, I guess Iāll take an enforced break from the game (even though I just came back).
Hey Dodgy,
From what I can see, you did receive the dragonriding mount already. You can find them at the top of your mount list. 
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I have the same problem as @Dodgy above. The common denominator here is that we acquired Dragonriding on MoP Remix before playin Dragonflight at all.
We have the mount available and can find it on or mount list, hoever we canāt actually use it - we do not have the Dragonriding skills available.
Here is the same bug report from US side: https://us.forums.blizzard.com/en/wow/t/dragonriding-not-working-on-standard-characters/1877904/1
This is incredibly frustrating because we cannot progress in the game at all.
Can you find a way for us to get the Dragonriding skills on our spellbook?
Okay Iāve worked it out and leaving my solution in case it works for others: Despite having Dragonriding in my spellbook and the corresponding mount unlocked through MoP Remix, you wonāt be able to use Dragonriding at all until you get the achievement āA new Friendā. To get this achievemnt you need to search your bags for the Renewed Proto-Drake mount. Unlock that mount and your existing Dragonriding mount from MoP Remix will start working as expected.
Iāve got 2 friends that havenāt done DF content but are currently going through remix. Thank you for taking the time to put a solution if the issue comes up with them.