TL;DR: DPS her immediately, click fetters every 3%, she dies after 27 successful fetters clicks when she hits 19%. Mezz adds, kill unmezzable.
Each of the waves of adds can mostly be mezzed. Only the named mobs seem to need to be killed. The non-named adds will despawn after a time - maybe 30s or so? The named mobs may also despawn at the same time if not killed, or perhaps the non-named mobs (yellow con if doing this in-era) despawn a certain amount of time after the named is killed. I'll try to test at next week's raid to see if they can all be kited until despawning, but with the AoE stun that might be a tall order.
When Ayonae Ro performs her "vulnerable" emote, one person should click their fetters. Note that the fetters must be equipped to click, and she must be targeted. Easiest tactic is to have one person sort the raid tool alphabetically and call out who's next to click and who's up after them. Emote happens, "Soandso, click. Good job, Someoneelse next, Thirdperson on deck." Missed fetters clicks are a big deal, as it has a whopping twenty minute cooldown.
Ayonae Ro's HP is gated behind fetters clicks - every time she becomes vulnerable and is successfully clicked, she can be damaged another 3%. She doesn't seem to become vulnerable again until she reaches that threshold. Every 3% someone will click their fetters then you'll DPS her another 3%. Repeat this 27 times, allowing you to damage her down to 19%. I believe she just dies at this point, though there may be one final click of fetters needed. I didn't have a 28th entry in my logs but the person who clicked might have been too far from me.
The "A look of utter fear crosses Ayonae Ro's visage as she is trapped by _____'s unseen fetter. She writhes and struggles but is unable to free herself from the bonds." line never happened. This may be from an older version of the event where you need to click her 30 times before she can be damaged.
Information from the Aradune server, PoR era.