The Call of Fate

Once one of your characters has chosen a Covenant, this quest will pop up on your alts after you've done the Shadowlands introductory experience in the Maw. You should get it as soon as you log in. If not, go to Oribos and relog.
Once you get it, go to Fatescribe Roh-Tahl in the back of the inn in Oribos. Go through the dialogue and click "I am ready to choose my fate in the Shadowlands." Finally, you're faced with two choices:

"Threads of Fate" completely locks you out of doing the main story missions that you originally did on your prior character. Instead, you will be given the quest "Re-Introductions" to immediately go to Tal-Inara to choose your Covenant. After that is "Aiding the Shadowlands" where you pick a zone (not related to your chosen Covenant), meet the leader (who notably treats with considerably less respect than if you had gone through the storyline. You don't even go through the cutscenes of getting your powers, you just start with them) and begin getting exp by world quests, certain world objectives, and dungeons.

"Replay the Storyline" is exactly what it says on the tin: You simply go through the story from Bastion all the way to Revendreth as you did before.

EDIT: Some more details. If you decide at any time that you want to stop doing Replay the Storyline, you can return to Roh-Tahl and click on Threads of Fate. However, the inverse is not true. If you are tired of Threads of Fate and want to return to Replay the Storyline, you can't, you are stuck in that timeline, so choose carefully.
In ToF, it only locks you out of the MAIN STORY quests. All other quests in the zone can be complete. Also keep in mind, ToF gives out gear far less often and reliably than the main story quests do, meaning that even regular World Quests will be notably tougher since you won't have as strong gear as you'd usually have.

Is this quest even available anymore? I can't figure out how to acquire it.

According to the following blizzard support article (only 4 months old as of this post), this quest is no longer available:

https://us.battle.net/support/en/article/277497

The Threads of Fate Campaign skip in Shadowlands is no longer available for leveling or max-level characters.

Any characters who have already chosen Threads of Fate prior to Dragonflight will be able to finish it. However, characters who never initiated Threads of Fate will be restricted only to the zone-by-zone Covenant campaign for Shadowlands.

If you already chose Threads of Fate, do not choose Chromie Time's Shadowlands - Realms of Death. If you do switch to Realms of Death while Threads of Fate is active, you will retain most of the features of Threads of Fate, but you will lose access to World Quests until level 60.

This article is inherently false. I literally just got prompted w/ threads of fate on a fresh dracthyr upon arrival in Oribos for the first time. I got to choose my fate, my covenant and even got a skip for Zereth Mortis. However, my other max level characters did not get this option. I have completed the campaign on my main. My first level 70 alt didn't get the prompt at all. My 2nd level 70 alt SOMEHOW, acquired the quest "the call of fate" without ever getting the threads of fate follow up.

So I have a ticket in. They're probably going to shine me on. This doesn't even seem remotely fair or right, especially considering what their support article states.

Edit for Ticket response:

Greetings!
Thank you for contacting Blizzard's customer support, I'm Game Master Igtenawcue, here at your service!
I can see you're reaching out because you're not being able to use the Threads of Fate skip. Let's look into this!

After investigating your situation, I was able to see that the Threads of Fate Campaign skip in Shadowlands is no longer available for leveling or max-level characters.
Any characters who have already chosen Threads of Fate prior to Dragonflight will be able to finish it. However, characters who never initiated Threads of Fate will be restricted only to the zone-by-zone Covenant campaign for Shadowlands.
If you already chose Threads of Fate, do not choose Chromie Time's Shadowlands - Realms of Death. If you do switch to Realms of Death while Threads of Fate is active, you will retain most of the features of Threads of Fate, but you will lose access to World Quests until level 60.

If, however, this feels like it's not working as intended, we do kindly ask for you to submit a Bug Report so that the dedicated team can look further into this, as this is something that us at Customer Support can't address at this channel. You can do this either by going to the dedicated forums or using the in-game bug report feature.

Dedicated Bug Report Forum: https://us.forums.blizzard.com/en/wow/c/support/bug-report/17

To report a bug in-game:

- Press Esc or click the ? button to open the menu.
- Click Support.
- Click Submit feedback or bug report.
- Enter the details of the bug into the Submit a bug description box, making sure to follow the guidelines outlined on the submission form.
- Click Submit.

Have a great weekend!
Game Master Igtenawcue

So they literally quoted the article I mentioned to them in my ticket back to me, and closed the ticket, without answering my question why a fresh dracthyr had access to a removed questline.

The %^&*. Typical blizzard response. Wow.