The use of Prooftext

Today, we continue with Peter's speech on the Pentecost (Acts 2:33-41).
After proving, or rather bearing witness to, Jesus' resurrection, Peter continued to deal with Jesus' ascension.
For Jews, Jesus, the Galilean carpenter, was no more than a rabbi or a prophet. Muslims do not have to deal with the complexity of the two natures of Jesus because for them, Jesus was just one of the prophets. For Christians, Jesus is the Son of God. The Son of God had descended from heaven, took up the form of a servant to die on the cross to redeem mankind. When he had finished his mission, what better place it was for him to return to if not heaven? Jesus had completed his work. He appeared to a selected few and ascended to heaven in their sight, leaving no evidence behind, not even an empty tomb. Now, how would you tell a crowd of potential believers? Some were skeptics, but some would eventually believe.
Before Peter delivered his first Christian proclamation to the world, the Holy Spirit descended in strong winds, thunders and tongues of fires on the 120 believers who then began to speak in tongues. These events alerted the people in Jerusalem, pilgrims as well as residents. Peter had no admissible evidence to demonstrate the ascension of Jesus. He only had these eye-witnesses who were now speaking in tongues, possibly in trance.
Being therefore exalted at the right hand of God, and having received from the Father the promise of the Holy Spirit, he has poured out this which you see and hear (Acts 2:33).
Peter interpreted this speaking in tongues as a manifestation of the Holy Spirit in the believers. This descent of the Holy Spirit was possibly only when Christ had returned to heaven. Again, this is an interpretation and there is no hard evidence. Therefore, alternatives are possible.
Nowadays, there are many charismatic Christian groups in which such speaking in tongues is seen. I have never joined any and will most likely not. I do not know if they are conscious of the fact that they are bearing witness not just to the infusion of the Holy Spirit, but that the ascended Jesus sends the Holy Spirit to them. Probably they are not because they are in trance. A Peter is needed to interpret the phenomenon to the world and we know very well that any interpretation can never be conclusive and exclusive.
To reinforce his interpretation, Peter resorted to Psalm 110, a favourite piece of prooftext for Christians. Jesus used it before to challenge the scribes who taught that the Messiah would be a descendant of David. Jesus wanted them to understand that the Messiah is more than the descendant of David. The Messiah is the Son of God, God Himself. Here comes the messy complexity of the two natures of Jesus --- his humanity and his divinity. As a man, Jesus is the descendant of David. As God, Jesus is the Lord of David. This is the use Christians found in Psalm 110. It becomes their prooftext to show that Jesus is the Son of God.
Peter found a new use of this piece of prooftext. He used it to demonstrate that Jesus had ascended into heaven, sitting at the right hand of God. From the point of view of a non-believer, this piece of text proves nothing definite. It cannot provide an airtight proof that Jesus is the Son of God, nor that he has ascended into heaven. It rings truth only in the ears of the believers. Luke wrote this Luke-Acts corpus to assure the believers.
that you may know the truth concerning the things of which you have been informed (Luke 1:4).
I wonder how the Jews interpret Psalm 110. I would like to know the proper ways to use a piece of prooftext, and if there is any limitation of the scope of its application.

My Lord, I believe in the witness and the interpretation offered by St. Peter. I know that we are not far from the truth. We long for Your return. In the meantime, allow me to serve You in a more appropriate way. Amen.

Appendix:
The Lord said to my Lord, Sit at my right hand, till I make thy enemies a stool for thy feet.εἶπεν [ὁ] κύριος τῷ κυρίῳ μου· κάθου ἐκ δεξιῶν μου, ἕως ἂν θῶ τοὺς ἐχθρούς σου ὑποπόδιον τῶν ποδῶν σου. (Acts 2:34b-35, Psalm 109:1 LXX, 110:1 MT)