@little_Socrates
Neither of those texts have ever been used by Jews as referring to the Messiah.
The idea of a personal messiah grew only after the Roman occupation during the Second Temple, and then only among some Jews, most notably the Essenes. Prior to this period there was no clear personification of the Messiah as an actual individual, and even afterward the fall of the Second Temple the idea was often spread among two persons or illustrative of the Messianic Age itself. Many Jews today do not expect that an actual individual called the Messiah will come, understanding references to "him" as symbolic of redeemed humanity during Olam Ha Ba.
The texts you mention in Psalm 22 and Isaiah 53 have far different meanings in Judaism. The idea of a suffering Messiah was so unknown and alien to Jews that even among the apostles the idea that Jesus would suffer was considered ludicrous. Remember how Peter objected to Jesus' own statement of his coming passion?
Read Matthew 16.21-23. If the Jews had ever expected that the texts you mentioned would apply to the Messiah, why did Peter not believe Jesus that such a fate awaited him? At Matthew 16.16 Peter had just said that he believed that Jesus was the Messiah, but then in verse 22 Peter also adds that a destiny of suffering would "never" come upon the Messiah. Why? Because like all other Jews, Peter did not believe in a suffering Messiah. The idea came from Jesus, and only after his death did Jesus' followers begin to declare that such suffering was necessary, never before.
The text of Psalm 22 is a prayer used to this day when Jews personally face suffering. It recounts the experiences of David, but it never says these experiences would befall the Messiah. Isaiah 53 is also merely a poetic description of the Jews suffering as if they were one individual.
Until recently Christian Bibles used to render Psalm 22 quite dishonestly. This has been corrected in newer versions such as the NRSV and the NABRE. Verse 16 in Old Testament renderings used to follow the midrash of the New Testament writers: "They pierced my hands and my feet" meant to describe the nailing of Jesus' feet to the cross. But in reality the text (which in Hebrew is verse 17) reads: "Like lions they maul my hands and feet."
Similar peculiar wordings of so-called "Messianic prophecies" in the Old Testament based on incorrect quotes in the New Testament have been similarly returned to their more precise translations in these modern versions. Scholars admit that past translators have performed a disservice not merely to the Jewish text but to Christians by failing to provide them with accurate renderings of texts like these and others such as Isaiah 7.14.
Returning them to their original state has left many Christians uncomfortable, true, but others have been able to embrace the fact that midrash was involved in their application by followers of Jesus.
This means such texts as you mention are no longer seen as Messianic in and of themselves even by many modern Christian exegetes (in other words not originally composed as or viewed as having any connection to the original Jewish hopes regarding the Messiah), leaving them to find new ways to explain their use by New Testament writers.
In the end you have Jewish history, present Jewish religious thought, Biblical translation scholars, and even Christian exegetes who you will find formidable in their disagreement to your argument based on the texts you mention.