>Many Christians believe Jews are saved because they're God's chosen, whether or not they convert and accept Jesus Christ.
No. Dispensationalists might believe that. But they would be wrong, as the witness of orthodox Christianity (in all its many historical manifestations) from the resurrection until the 19th century affirmed.
Acts 4:8–12 (ESV): Then Peter, filled with the Holy Spirit, said to them, “Rulers of the people and elders, if we are being examined today concerning a good deed done to a crippled man, by what means this man has been healed, let it be known to all of you and to all the people of Israel that by the name of Jesus Christ of Nazareth, whom you crucified, whom God raised from the dead—by him this man is standing before you well. This Jesus is the stone that was rejected by you, the builders, which has become the cornerstone. And there is salvation in no one else, for there is no other name under heaven given among men by which we must be saved.”
Peter’s audience comprises Hebrews - including Temple hierarchy and rabbis.
Peter tells his audience that Jesus is the stone that was rejected that has become the cornerstone - an allusion to Psalm 118 - and that it is this crucified and risen Jesus who is the agent of the salvation that the royal/messianic psalm promises.
One example of dozens that could be adduced to show one way of salvation and one way only.