From what I gather, yes the Sharingan was simply a portion of the Sharinnegan power afforded to one of her offspring. Why this Sharingan evolved to Rinnegan with no tomoe isn't too blatantly obvious but one can assume it was more of a mutation of the eye than anything, meaning Kakashi was right about that idea.
The real problem is the difference between Kaguya and Madara's Sharinnegan and Sasuke's tomoe Rinnegan. From what we've seen, the sharinnegan does nothing but cast IT. Sasuke's tomoe Rinnegan only allows him to trade places with another object. In reality, those are both kind of lame considering the massive capabilities of normal Rinnegan, including the use of Limbo clones. Why was Sasuke's tomoe Rinngan purple when the sharinnegan is red? Its all a bit sketchy.
They are the same eye, but one is a more powerful mutated version of the other. The Rinnegan DOES have an upgraded version of the Sharingan's copying technique, that being gaining an affinity for all elements. A sharingan can copy jutsu, but only single elements that are within the user's affinity. The Rinnegan allows a user to learn ANY jutsu from ANY affinity. I say that's a superior version of the same concept.
The major thing to realize about this is Oro's plan. He wanted the sharingan mainly to learn new jutsu, as that was his ultimate goal. Yet, after the war, we learned Oro's true ultimate goal was to unlock Rinnegan from sharingan, which would have secured his chance to learn every jutsu in the world. He'd still be limited by elemental KKG, but without a doubt having Rinnegan would achieve his goal and that all-affinity capability is the ultimate form of the Sharingan copy ability.