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I agree that the beast tribe FATEs are annoying, but you can force them to pop by completing other FATEs linked to them. This really only applies to the Sylphs and Amalja, since the other two thankfully pop frequently.
I know that to force pop "What a Boar" you should complete "Everythings Better" and "Breaking Dawn". I've done this a dozen times after I read up on the connected FATE, works like a charm.
For "Quenching the Flame" I know you need to complete "Heroes of the 2nd". I'm unsure if you need another FATE or not though.
EDIT: It seems that "Taken" is likely the third FATE you need, as it also is 10 minutes long.
Its not a solution, but it helps pops the FATE that you need.
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The FATE beast tribe quests should just be REMOVED ENTIRELY. No complicated spawn clocks, no special trigger items, etc. These are band-aids on the main underlying issue, which is that when you are asking players to do -daily- quests, expecting players to stand around camping FATEs is preposterous.
The end.