The need for Christ's death on the cross is not questioned because it is indeed necessary if penal substitution and the surrounding doctrines are true. I do not believe this theory is taught by the New Testament. Once we look beyond this theory, though, we can notice the obvious - Jesus died as a martyr. He was put to death by authorities who did not like his teachings and actions. Jesus taught a message of social and cultural revolution. He gathered a following to further this revolution so that people could enjoy the benefits of experiencing this new kind of life that he called being in the "Kingdom of Heaven". Jesus brought people into the Kingdom of Heaven, and that is why he was killed.
The cross, then, is naturally a symbol of Jesus' life and message. A powerful symbol, but a symbol nonetheless. The apostle Paul referred to the cross in this way to encapsulate Jesus and his message. For early followers of Jesus it would have been obvious to Christians that his death was not significant because of some atonement theory only voiced centuries after Jesus. Rather, the cross is significant because of his life and teachings. Likewise, his resurrection is important because of both his life and his crucification.
Christians don't like this idea because it means that the cross was not needed to fulfill the Great Spiritual act of Atonement through penal substitution. They might think that this makes the cross meaningless. But they would be missing the message encapsulated and symbolized by the cross.
Jesus used the cross as a symbol of what it meant to follow him:
If anyone desires to come after Me, let him deny himself, and let him bear his cross, and let him follow Me. (Mat 16:24)
Go, sell what things you have, and give to the poor. And you will have treasure in Heaven. And come, follow Me, taking up the cross. (Mar 10:21b)
Whoever does not bear his cross and come after Me, he cannot be My disciple. (Luk 14:27)
St. Paul used a mixture of analogies and symbolism to discuss Jesus and his message. He too used the cross to symbolise the Way of life Jesus lived and encouraged:
Be fellow-imitators of me, brothers, and consider those walking this way, even as you have us for a pattern. For many walk as hostile to the cross of Christ... (Php 3:17-18a)
Jesus often opposed the Torah as it was being practiced, and Paul appears to have followed Jesus' teachings. Those faithful to Jesus' teachings were thus often persecuted by Jews who wanted to enforce Torah (such as the Pharisees) because they saw the "offense of the Cross" (Gal 5:11). Paul refers to their persecution by saying they are "persecuted for the cross of Christ" (Gal 6:12). Then he says that he does not boast in the Torah, but in the "cross of our Lord Jesus Christ" (Gal 6:14) in which circumcision makes no difference (verse 15).
In Ephesians he describes how the message of Jesus overcomes cultural and social barriers, thus uniting people in a new way within the culture of the "Kingdom of God". Specifically he refers to it breaking down the division between Jews and Gentiles that comes from the radical new values Jesus taught. Paul uses powerful, emotive symbols of both the cross and Jesus' blood (which also carries connotations of kinship) to encapsulate the message of Jesus. Perhaps he is referring also or instead to the fact that without Christ's martyrdom and subsequent resurrection Christianity probably wouldn't have got started. Listen to what (I think) Paul writes:
But now, in Christ Jesus you [Gentiles] who then were afar off, came to be near [to us Jews] by the blood of Christ. For He is our peace, He making us both one, and breaking down the middle wall of partition, in His flesh causing to cease the enmity [created by the commandments of the Torah], that He might make these two groups united into one new man - his "body", making peace, and that He might reconcile both in one body to God through the cross, slaying the enmity [within His body]. (Eph 2:13-16*, see also Col 1:20)
But Paul doesn't refer to the cross in only this way. He uses it's significance in many ways to make many different points. For example, like the Gospel and other New Testament writers, notes the shame of his death on a cross (Php 2:8). In another passage, Paul uses the imagery of a list of things against us being nailed to the cross (Col 2:14). Paul's point when referring to the cross is not always the same. Combined with penal substitution, Paul's use of many different images and metaphors can make it hard to recognize that the cross found its significance because of Jesus' life and teachings.
In summary, the question "Why did Jesus need to die on the cross?" is based in a number of presupposed theological ideas that don't help us understand why he did die on the cross, and what his death came to symbolize for his followers. His death is significant because of his life. And while there was no "need" for him to die on the cross, without his crucifixion and resurrection, his following would have probably died instead. Without the cross, we might not even know who Jesus was. But that doesn't mean the cross was needed for some Great Spiritual Act of Atonement, nor is it what the cross means. The cross represents what Jesus lived, taught, and was ultimately killed for.
* In quoting Eph 2:13-16 I have intentionally changed the phrases "in Himself" to reflect that I think this phrase refers to people who are "in Christ" - and thereby in his group, his disciples, his "body".