why did Peter and the Jerusalem apostles lose their authority as Israel rejected the Messiah?
Because Israel continued rejecting Jesus as Messiah and His coming kingship, Peter and the Jerusalem apostles' authority diminished. This was not Peter's fault; he preached faithfully. As Israel slipped into unbelief, the whole system fell through the cracks, and God turned to the Gentiles through Paul's ministry.
Les explains that Peter and the Jerusalem apostles had been promoting Christ as Israel's Messiah, His coming kingship, and His Kingdom, but Israel rejected Him. Les says this was not Peter's fault, for he had done all he could and preached his heart out in the early chapters of Acts. Israel could not believe that anything good came out of Nazareth. In God's divine purposes, as Israel kept slipping away in unbelief, Peter, the eleven, and the Jewish leaders of Jerusalem were losing their clout; the whole system was falling through the cracks. God was not out of it. As Israel slipped and rejected all this, God turned to the Gentiles. As the Jews went down, the Gentiles came out, which is Paul's ministry. In verse 6, those who seemed to be somewhat were the twelve and the elders at Jerusalem, but they no longer had the power rightfully as God had entrusted it to them.