I've allready read tons of advise on this forum, the official guide etc. And I've burned a lot of wood in the fire with this tribe. The current fire has been lit about three weeks now and I'm 100% sure it hasn't gone out at any point during the that time. I've been patient.
You HAVE let the fire go out, right? The key doesn't appear if it doesn't go out. I've heard of other people, though, who haven't had sparkles, but still found the gem when the fire went out.
Couple of weeks ago I started four other tribes to see if running more tribes would increase the possibility to see all that talk in FAQs and forums is not just lies upon lies.
Increase the probability of what? There is no percentage chance involved here to the best of my knowledge.
Is there a certain point in the game after which you can't get it anymore? Are there still some special conditions for this one, for example "the fire has to be lit for 65 hours or more for each hut/ 15 villagers/ tons of fish collected /etc you have in game"?
No. It's most likely a glitched timer in the game, possibly related to alterations in your system clock, possibly a bad save, possibly anything.
Try letting the fire go out, restarting the fire, and going for the sparkles again, from scratch. Keeping it lit for three weeks straight doesn't seem to be working; letting it go out once may reset the timer.
This puzzle is horrible. It's not too hard to solve, but even if I got instructions to do it step by step there is no way I can controll wether I'm doing progress here or not. All I can do is wait and feed the fire without any guarantee that doing this will pay for my efforts some day in the future. It's not entertaining at all if that's what the game designers thought.
If everything you said is true, the game is obviously not functioning as intended. After all, you said four other villages doing the same thing worked, right? The bad attitude towards the developers who 'designed' such a horrible and difficult puzzle is completely misplaced.