The Jewish leader leaders forced Paul to go to the temple to prove his Jewish credentials. This was not voluntary. He did it under duress. It clearly was a sop to the Christ-rejecting Jews.
There was a religious / political aspect to the sacrifices that all Israelis were bound to as Jews in order to maintain their Jewish privileges. I believe Paul yielded to the political requirements - in honouring the God-ordained authorities - but he made no spiritual submission to these abolished rites. I don't believe he recognised the expiatory significance of the sacrifices, as he finally released himself from the religious Judaic baggage he was bound with. He would not have recognised the intercession of the high priest with God for the people because he had a real, perfect and eternal great High Priest (Hebrews 5:6, 9, Hebrews 4:14).
Jennings' Jewish Antiguities, p. 17 says of Acts 21:26: "there was a political as well as a typical use of sacrifices; and that, though the typical ceased upon the sacrifice of Christ, yet the political continued until God in his providence broke up the Jewish state and polity about forty years after our Savior's death. Till that time it was not merely lawful, but matter of duty, for good subjects to pay the dues which were appointed by law for the support of the government and magistracy. Now, of this kind was the sacrifice which Paul offered; and in this view they were paid by Christians dwelling in Judea, as well as by those who still adhered to the Jewish religion. So that, upon the whole, this action, for which Paul has been so much censured, probably amounts to nothing more than paying the tribute due to the magistrate by law, which the apostle enjoins upon all other Christians in all other nations, Romans 13:6."
We do not know the motive for Paul going into the temple at this time, or if he in fact actually made a blood offering. It could have been legally required to maintain his citizenship. It could thus have been a sop to the religious authorities, albeit not in any way he believed that there was any spiritual purpose or efficacy in the rite. Of course there is the possibility that it was an imprudent compromise on Paul’s part. He may have meant well, participating in this solely to get an opportunity to preach in the temple. We just don’t know. However, we definitely cannot build a doctrine upon silence, especially when the New Testament is very clear that Christ’s sacrifice was the final sacrifice for sin forever.
The one thing we do know that not long after this incident Christ eradicated the whole Judaic sacrifice system through the destruction of the temple, proving that this system was over forever. It was impossible for this system to survive the destruction of its ceremonial centre of operations. This reinforces the belief that this arrangement was now considered rebellious, pointless and redundant with the arrival of the new covenant and Christ’s final sacrifice for sin.
Please list where Paul (or any other disciples) partook in animal sacrifices after conversion? Every time they went to the temple it was for the sole purpose of presenting Christ and exposing the error of the then Christless Crossless Jewish religion. There is nowhere in Acts 21 that suggests Paul engaged in animal sacrifices or the abolished rites. Many Dispy theories are based upon pure speculation. In fact, it doesn't even mention the feast that he attended in the chapter. Even if Paul attended a festival, it was for the sole purpose of preaching the Cross.