Other biblical accounts
You ask "Why was Aaron not punished?" Indeed from Deuteronomy 9:20 it seems that God was very upset with Aaron for making the calf and wanted to punish him (destroy him), but Moses besought god to spare him and that's how he was saved
And the Lord was angry enough with Aaron to destroy him, but at that
time I prayed for Aaron too.
But since it would be un-scholarly to conflate these two accounts and assume that they both agree with each other I will not take this route. Indeed, from Exodus 32 there is no hint whatsoever that God was furious with Aaron (only that Moses his brother was very upset with him) or that he intended to punish him. His sons are burnt alive, and he is barred from entering the promised land, but there is no indication in the text that these were intended as a punishment for Aaron for his involvement with the golden calf. So your question is valid.
I think the answer to your question can be found if we study the narrative closely. From studying the beginning verses it becomes readily apparent that the Israelites were not asking Aaron to make them an idol, or a new image to replace the worship of Yahweh, they were looking for a substitute for Moses. This is the basic outline: Moses does not return and the Israelites panic, they think he is lost or dead. They are desperate for a new leader to show them the way out of the vast barren desert, so they approach Aaron to build them an oracle, a cultic object or image that would act as a medium between them and God; they would ask questions and the calf, acting as a medium between them and Yahweh, would provide the answers. There was nothing wrong with making such an object, indeed David's household owned similar items and images (which were used as oracles) which were called Teraphim in biblical times. They were regularly used throughout Israel and weren't outlawed until much later during king Josiah's reforms (2 kings 23:24).
That the Israelites were asking for a substitute for Moses is evident from the beginning of chapter 32
When the people saw that Moses was so long in coming down from the
mountain, they gathered around Aaron and said, “Come, make us Elohim
who will go before us. As for this fellow Moses who brought us up out
of Egypt, we don’t know what has happened to him.”
The Israelites surely did not worship Moses, he was just a prophet acting as a medium between them and Yahweh, so when they were looking for a substitution for Moses they were looking for an object that would do just that: mediate between them and Yahweh. Or as Philpot eloquently puts it (p. 45), "The request, then, is not initially for another Yhwh, but for an image representing Yhwh in the place of Moses." (see Philpot's footnote for a bibliography of sources). The medieval Jewish commentator Rashbam has suggested a very similar approach.
(As for the word "Elohim" [אלהים], in my view it is better interpreted as as a "divine being" or "divine medium" than "gods" as most translations have them and in my view are erroneous. Indeed אלהים is elsewhere interpreted similarly. see Deuteronomy 33:1; Judges 13:6 where it is rendered "divine man".)
But when Aaron finished the golden calf some came to believe that it is not merely an oracle but an image of a god that took them out of Egypt, and they said "These are your gods,[b] Israel, who brought you up out of Egypt.” Whether they identified this god with Yahweh or some other deity is a matter of considerable controversy, but one thing is clear that at this point they started worshiping it, bowing down to it and sacrificing to it (see verse 6) and that's when God got angry with them. Aaron however was not part of this cult and had nothing to do with it, the people completely misinterpreted the image of the calf and took it to the next level when originally it wasn't built for this purpose, hence the answer to your question: Aaron was indeed innocent!
This sheds light on Aaron's answer to Moses, when the latter accuses him of leading the people to sin we find Aaron's reply,
Do not be angry, my lord,” Aaron answered. “You know how prone these
people are to evil. 23 They said to me, ‘Make us Elohim who will go
before us. As for this fellow Moses who brought us up out of Egypt, we
don’t know what has happened to him. So I told them, ‘Whoever has any gold jewelry, take it off.’ Then they gave me the gold, and I threw it into the fire, and out came this calf!”
Lots have attempted to clarify this dialogue, especially Aaron's reply which is ambiguous, but according to my understanding his answer is simple. Aaron is defending himself by pointing out that he only intended to make them an object that would act as a medium between them and god, but they mistook it as a god and started worshiping it; in other words, he had no part in this crime and thus emerges clean.