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For starters, a Jew would recognize in Peter’s sermon that Jesus died and was victorious over death in His resurrection.

They were steeped in the word of God, that death and corruption of the body entered the world through sin and all of their sacrifices were for the remission of sins, sin offerings.

23 You delivered him to sinners to be crucified and killed, and in this way the purpose of God from all times was fulfilled. 24 But God raised him to life and released him from the pain of death, because it was impossible for him to be held in the power of death

29 Friends, I don’t need to prove that the patriarch David died and was buried; his tomb is with us to this day. 30 But he knew that God had sworn to him that one of his descendants would sit upon his throne and, 31 as he was a prophet, he foresaw and spoke of the resurrection of the Messiah. So he said that he would not be left in the region of the dead, nor would his body experience corruption.
32 This Messiah is Jesus and we are all witnesses that God raised him to life. 33 He has been exalted at God’s right side and the Father has entrusted the Holy Spirit to him; this Spirit he has just poured upon us as you now see and hear.
34 And look: David did not ascend into heaven, but he himself said: The Lord said to my Lord: sit at my right side 35 until I make your enemies a stool for your feet. 36 Let Israel then know for sure that God has made Lord and Christ this Jesus whom you crucified.”
37 When they heard this, they were deeply troubled. And they asked Peter and the other apostles, “What shall we do, brothers?”
38 Peter answered: “Each of you must repent and be baptized in the name of Jesus Christ, so that your sins may be forgiven. Then you will receive the gift of the Holy Spirit. 39 For the promise of God was made to you and your children, and to all those from afar whom our God may call.”

You read this and do not see in it that Jesus died for their sins? Okay, that explains a lot.