So, I've been reflecting on stuff from both shows, and I've come up with a working theory for the origins of Bending and its relation to Spirits.
In A:tLA, we learn that the original Benders learned straight from the source: Firebenders from Dragons, Airbenders from Sky Bison, Earthbenders from Badgermoles, and Waterbenders from the Moon and Ocean. In Beginnings, we see how Wan became the first Avatar by directly acquiring Bending from the Lion Turtles, long before civilizations like the Sun Warriors or Oma and Shu's people and seemingly contradicting stuff in the Last Airbender
From what we know, Raava and Vaatu are two of the most powerful spirits in either world, with the latter being responsible for tearing open the Spirit Portals in the first place and the former being the Avatar Spirit. They are of a higher order of spirits, like Tui & La, Koh, the Mother of Faces, etc. They're the big movers and shakers of the Avatar world.
My theory is based on the fact that Tui and La, the Moon and Ocean (and thus, the origins of Waterbending) were spirits who chose to take physical form in our world. What if the original sources of the other three Bending arts (Bison, Dragons, and Badgermoles) were, themselves, spirits? Perhaps of a lesser order than the likes of Raava and Vaatu, and more numerous than Tui and La. Maybe when Wan closed the Spirit Portals and most spirits returned to the Spirit World, some spirits took physical form as the three animal species, and their power manifested as the ability to manipulate fire, earth, and air. And maybe the Lion Turtles were of a higher order, perhaps closer to the likes of Raava and Vaatu, who did this, and that would explain their enormous energy manipulation powers.
I like to think this would help to rectify the problem of the origins of Bending in relation to Beginnings, but what do you guys think?
I think it worked that way:
The people got the power to bend from the lion turtles. Years later, maybe centuries, they learned how to control the bending in the right way from the moon/badgermoles/dragons/sky bisons. Maybe the story in the cave of the lovers was just metaphoric, so Oma and Shu weren't really the first earthbenders, but the first ones who knew how to bend, how to handle with their power. And this was what made them so strong to end a fight between two cities which endured many generations. The Sun Warriors did it the same way, they were probably just much better firebenders than the other ones, but not the first or the only ones at the time.
I don't know how dragons/sky bisons/badgermoles learned how to bend, but they could have been given the bending power from lion turtles, too. Tui and La are high spirits, so they have never had problems with pulling water up and down.