First, the Creed does not say Pilate killed Jesus. As you point out in your question, there was plenty of blame to go around for Jesus' death. Biblically, we (i.e., you and me and every person who ever lived) are partly to blame. Let's also not forget that God the Father
. . . did not spare His own Son, but delivered Him up for us all . . ..
This delivering up by God the Father was planned from eternity past in the counsels of the Triune God. Nothing could stop it from happening, and no single person or group of people should shoulder the blame. While in a sense all humankind is culpable before God, God's plan was always to put forward his Son as the "Lamb of God who takes away the sin of the world" (John 1:29 & 36).
When Jesus said from the cross, in essence,
Father, forgive them, for they are acting in ignorance,
the word they included not only the political- and the Jewish religious authorities of Jesus' day, but it includes all of humankind.
Second, in some versions of the Nicene Creed, the word suffered has been changed to the word crucified.
For example, the ecumenical version of the Creed which was published by the ICET (International Consultation on English Texts) is worded this way:
For our sake he was crucified under Pontius Pilate.
While the Scripture does not in so many words tell us why God allowed Jesus to be tried before Pilate, one possible reason could have been to demonstrate Jesus' utter submission to the Father's will, even if that submission involved coming under the authority of an earthly potentate (though Pilate was a mere governor of a province, or district, within a very small country). As Paul pointed out in his letters to the churches (in Romans, for example),
. . . for there is no authority except from God, and those which exist are established by God (13:1).
Jesus was not a law breaker, which fact alone made his trial a sham. Recall the occasion during his public ministry in which Jesus provided a coin miraculously for his payment of the temple tax. Even he "rendered unto Caesar the things which are Caesar's"! (Matthew 22:21; Mark 12:17; and Luke 20:25).
The central irony, of course, is that the "King of kings and the Lord of lords" allowed himself to be tried by a lesser, second-rate lord in order to fulfill God's plan for the ages.
Another possible reason for the inclusion of Pontius Pilate's name in the Creed was simple to ground in history the events surrounding Passion Week. Jesus' trial and his death, burial, and resurrection happened in a specific historical, religious, and cultural context, as did the other events in his life. Paul put it this way,
But when the fullness of the time came, God sent forth His Son, born of a woman, born under the Law, so that He might redeem those who were under the Law, that we might receive the adoption as sons (Galatians 4:4-5 NASB).
For years, theologians and Bible students have speculated on the reasons why God considered the first century of the Common Era to be the "fullness of time" (or, "the appropriate time," which is how the NET translates the Greek idiom).
They cite numerous historical and cultural factors which made Jesus' appearance in time and space to be the most propitious time, factors such as the Pax Romana; the existence of Roman-built roads which linked disparate parts of the Empire, making the spread of the gospel easier; the Roman admiration for Greek culture, particularly their emulation of the Greek literary tradition which included many forms of writing, including letters (or "epistles"), which comprised the bulk of the New Testament (and in koine Greek!); and the authoritative organization and hierarchical structure of Roman government, which made such things as "worldwide taxation (or "census," for assessing property taxes) possible, a factor which brought Joseph and Mary to Bethlehem! (The foregoing factors comprise a very short list indeed.)
The Gospels make perfectly clear that Pilate was not the only Roman authority implicated in the death of our Lord Jesus. Herod Antipas interviewed Jesus but he "passed the buck" and wound up sending him back to Pilate. Pilate also "washed his hands" of the Jesus affair and gave in to the will of the mob. Pilate may also have been influenced by the mob's specious argument that for him to release Jesus would mean Pilate was "no friend of Caesar [Augustus] (John 19:12).
In conclusion, Jesus' crucifixion under the authority of Pontius Pilate was simply another way for God to bring His will to pass. Pilate was a tool in the hands of God. For all eternity, God the Father and His well beloved Son knew what was going to happen, when and where it would happen, and even gave promises, hints, shadows, and types to believers who lived millennia before Jesus was born. After he was born, only gradually did his followers begin to realize (and did Cleopas and the unnamed disciple in Luke 24) that all the major events of Jesus' earthly life occurred according to "the predetermined plan and foreknowledge of God" (Acts 2:23 NASB).