In order for David and Bathsheba’s son to have been the long-awaited Messiah (or Christ) he would have had to have fulfilled some 300 prophecies before he died. Clearly, he did not. The whole Old Testament—Law, Prophets, and Writings—contains messianic prophecies. Here is an extract from an article that shows how those prophecies were not fulfilled till Jesus was born:
Jesus’ birth is the fulfillment of Isaiah 7:14 (cp. Matthew 1:18–23). Jesus’ flight to Egypt turns out to be the fulfillment of an indirect prophecy in Hosea 11:1 (cp. Matthew 2:15). Jesus’ triumphal entry into Jerusalem is linked to Zechariah 9:9 (cp. Matthew 21:1–5). Jesus’ death on the cross fulfilled many Old Testament prophecies, including Psalm 34:20 and Zechariah 12:10 (cp. John 19:31–37).
At times Jesus quoted a messianic prophecy and applied it to Himself. In the synagogue in Nazareth, Jesus read a messianic passage from Isaiah 61 and said, “Today this scripture is fulfilled in your hearing” (Luke 4:21). Just before His arrest, Jesus quotes Zechariah 13:7, stating that prophecy is about to be fulfilled (Matthew 26:31). He also quotes Isaiah 53:12 (in Luke 22:37), and when we study the whole of Isaiah 53, we discover that much of that chapter corresponds directly to Jesus’ passion. When Jesus quotes an Old Testament passage and says that He is the fulfillment of it, we know for sure that passage was messianic.
Sometimes Jesus’ allusion to a passage clues us in that we’re dealing with a messianic prophecy. For example, on the cross Jesus cried out, “My God, my God, why have you forsaken me?” (Matthew 27:46). As it turns out, those are the exact words of Psalm 22:1. When we turn to Psalm 22, we find many details of the crucifixion: the mocking Jesus endured (Psalm 22:7; cp. Matthew 27:38–44), Jesus’ thirst (Psalm 22:14; cp. John 19:28), the piercing of His hands and feet (Psalm 22:16; cp. John 20:20), and the casting of lots for His garment (Psalm 22:18; cp. Luke 23:34). Jesus’ agonized cry serves as a signpost pointing us to a treasure trove of messianic prophecies in the Psalms. Source: https://www.gotquestions.org/messianic-prophecies.html
For a more detailed list of Messianic prophecies, which were fulfilled in Jesus, please read this article: https://www.gotquestions.org/prophecies-of-Jesus.html
With regard to the two genealogies of Jesus, please consider this:
Most conservative Bible scholars today take a different view, namely, that Luke is recording Mary’s genealogy and Matthew is recording Joseph’s. Matthew is following the line of Joseph (Jesus’ legal father), through David’s son Solomon, while Luke is following the line of Mary (Jesus’ blood relative), through David’s son Nathan. Since there was no specific Koine Greek word for “son-in-law,” Joseph was called the “son of Heli” by marriage to Mary, Heli’s daughter. Through either Mary’s or Joseph’s line, Jesus is a descendant of David and therefore eligible to be the Messiah. Tracing a genealogy through the mother’s side is unusual, but so was the virgin birth. Luke’s explanation is that Jesus was the son of Joseph, “so it was thought” (Luke 3:23). Source: https://www.gotquestions.org/Jesus-genealogy.html
As pointed out by Nigel J, “None of this agrees with the writings of the apostles of Jesus Christ. Nor does it agree with the prophecies of the prophets of Israel. It is entirely opinion-based.”