According to Roku in ATLA the past lives and experiences come through during the avatar state? Why then did Aang have to master the four elements if his past lives theoretically could use them for him in the avatar state? Or is this an incorrect assumption?
Cost of the using the Avatar State
Using the Avatar State is a powerful tool, but comes at equal cost. Remember that Aang consumed a lot of his life force after preserving himself in an iceberg for 100 years (using the Avatar State). It can also be seen that Aang is visibly drained after his usage of the Avatar state for prolonged periods of time. In season 2, episode 1 of A:TLA, Roku speaks to Aang and states that
"In the Avatar State, you're at your most powerful... but you are also at your most vulnerable."
So depending on the situation, Aang risked ending the Avatar cycle completely, as Roku also states that
"If you are killed in the Avatar State, the reincarnation cycle will be broken and the Avatar will cease to exist."
Cost summary: It was more physically demanding and risky to use the Avatar State as your only source of bending power.
Bending Strength and spirituality
Note: Some of the points in this section may be seen as primarily opinion based. For instance. To explain my reasoning on these, I will include brief notations.
One major points of emphasis for Avatar Korra is her lack of spiritual sensitivity. She is eventually able to master these whilst mastering her remaining bending technique: air. She was overly aggressive and eventually the adoption of air-bending ideologies helped to balance her. Had Aang forgone mastery of all of the bending techniques, he wouldn't have mastered his spirituality. He would probably still be peaceful and enlightened, but would've lacked the discipline and head-on attitude that earth bending required out of him. I summarize this point by saying that balance was necessary for an Avatar to achieve. Mastering all of the elements helped create a balance within each Avatar.
It is suggested through various interactions in both The Legend of Korra and A:TLA that your own ability as a bender was enhanced by your past lives. Thus, you would be at a disadvantage had you not fully mastered each elements. Take Avatar Aang's many experiences in the Avatar State for example: He is generally shown to lack control and full rational-thinking whilst in the Avatar State, which results in very dangerous behavior. So even though he channeled the power of his past lives, he lacked control over those elements due to his lack of spiritual training and bending mastery (this is explained in Season 2, Episode 19 "The Guru") . To summarize this point, It is evident that control over the avatar state and bending in general was reliant on your mastery and spirituality, which would be absent without the later. Here is a quote (again, from Roku) that briefly explains the strength of the avatar state:
"The glow is the combination of all your past lives, focusing their energy through your body."
- The Avatar State was a costly tool that was both physically and mentally draining
- Aang, being the Avatar and all, needed balance to achieve his purpose. Had he not struck a spiritual balance through the mastery of his bending, he would not be effective in gifting the world balance it required from him.
- The Avatar State was never shown or meant to be the end-all bending tool. It was to be used in moments that required great strength from the wielder ( EX: Aang raising the oceans to extinguish the fires caused by Sozins comet).
- To say the least, without learning to control the Avatar state as he could at the end of Aang's journey in A:TLA, the tool would make him an uncontrollable and unstable bending-ball of chaos.
There are some more reasons, but I felt this provided an adequate answer to your question.
To my previous point of the Avatar dying at a young age, we do indeed know that many known figures in A:TLA lived very long lifespans, so there should be no discrepancy. Sixty Six is still not the longest of lifespans.