I don’t understand what you’re saying. Are you saying that to convince these Jews that Jesus was the Christ, Paul explained to them the kingdom Jesus said was at hand but didn’t establish?
He showed them out of the Law and the Prophets that He Whom they had crucified had been the Christ.
He attempts to get that settled.
He did that in Acts 17, for example.
IF he is allowed to get that settled in their minds, he THEN proceeds to lay out that Daniel's 70th Week has been interrupted.
You see his having done just that in Acts 17 when you read his reminding them of this in his writings to them - 1st and 2nd Thessalonians.
In 2nd Thessalonians 2, for example, he reminds them he had taught about said delay "when I was yet with you."
He lays all that out WHEN speaking to JEWs and they receive the first part of THEIR equation - that Jesus had been their PROPHESIED Christ.
In 1st and 2 Thessalonians he also reminds them of his having taught them what part of God's Kingdom they are a part of.
That they have been delivered from that wrath to come in which the Lord returns to restore David's throne on the Earth. They are not part of that part of God's kingdom.
Those who will be a part of that, will be going through its Tribulation FIRST, see Peter's preaching to Israel, in Acts 3.
When Paul meets with unbelief on the part of Jews, he does not go into all that.
Instead, he moves on after relating he that "lo" he will go to the Gentiles - that "they will hear it."
Acts 18:4 And he reasoned in the synagogue every sabbath, and persuaded the Jews and the Greeks. 18:5 And when Silas and Timotheus were come from Macedonia, Paul was pressed in the spirit, and testified to the Jews that Jesus was Christ. 18:6 And when they opposed themselves, and blasphemed, he shook his raiment, and said unto them, Your blood be upon your own heads; I am clean: from henceforth I will go unto the Gentiles.
As for the "church" in Matthew 16, that "church" is Messianic. Look at its required belief - "Thou art the Christ."
Matthew 16:13 When Jesus came into the coasts of Caesarea Philippi, he asked his disciples, saying, Whom do men say that I the Son of man am? 16:14 And they said, Some say that thou art John the Baptist: some, Elias; and others, Jeremias, or one of the prophets. 16:15 He saith unto them, But whom say ye that I am? 16:16 And Simon Peter answered and said, Thou art the Christ, the Son of the living God. 16:17 And Jesus answered and said unto him, Blessed art thou, Simon Barjona: for flesh and blood hath not revealed it unto thee, but my Father which is in heaven. 16:18 And I say also unto thee, That thou art Peter, and upon this rock I will build my church; and the gates of hell shall not prevail against it.
In John 1:47, that is tied to "thou art the Son of God...the King of Israel."
John 1:45 Philip findeth Nathanael, and saith unto him, We have found him, of whom Moses in the law, and the prophets, did write, Jesus of Nazareth, the son of Joseph. 1:49 Nathanael answered and saith unto him, Rabbi, thou art the Son of God; thou art the King of Israel.
This Messianic assembly is Matt. 21:43's "another nation."
It awaits the Lord's return to Israel, in His wrath.
Luke 12:31 But rather seek ye the kingdom of God; and all these things shall be added unto you. 12:32 Fear not, little flock; for it is your Father's good pleasure to give you the kingdom. 12:33 Sell that ye have, and give alms; provide yourselves bags which wax not old, a treasure in the heavens that faileth not, where no thief approacheth, neither moth corrupteth. 12:34 For where your treasure is, there will your heart be also. 12:35 Let your loins be girded about, and your lights burning; 12:36 And ye yourselves like unto men that wait for their lord, when he will return from the wedding; that when he cometh and knocketh, they may open unto him immediately. 12:37 Blessed are those servants, whom the lord when he cometh shall find watching: verily I say unto you, that he shall gird himself, and make them to sit down to meat, and will come forth and serve them. 12:38 And if he shall come in the second watch, or come in the third watch, and find them so, blessed are those servants. 12:39 And this know, that if the goodman of the house had known what hour the thief would come, he would have watched, and not have suffered his house to be broken through. 12:40 Be ye therefore ready also: for the Son of man cometh at an hour when ye think not.
Acts 1:6 When they therefore were come together, they asked of him, saying, Lord, wilt thou at this time restore again the kingdom to Israel? 1:7 And he said unto them, It is not for you to know the times or the seasons, which the Father hath put in his own power. 1:10 And while they looked stedfastly toward heaven as he went up, behold, two men stood by them in white apparel; 1:11 Which also said, Ye men of Galilee, why stand ye gazing up into heaven? this same Jesus, which is taken up from you into heaven, shall so come in like manner as ye have seen him go into heaven.
Notice verse 23 below - both Israel's are on the Earth when He returns - to clean house.
Acts 3:18 But those things, which God before had shewed by the mouth of all his prophets, that Christ should suffer, he hath so fulfilled. 3:19 Repent ye therefore, and be converted, that your sins may be blotted out, when the times of refreshing shall come from the presence of the Lord; 3:20 And he shall send Jesus Christ, which before was preached unto you: 3:21 Whom the heaven must receive until the times of restitution of all things, which God hath spoken by the mouth of all his holy prophets since the world began. 3:22 For Moses truly said unto the fathers, A prophet shall the Lord your God raise up unto you of your brethren, like unto me; him shall ye hear in all things whatsoever he shall say unto you. 3:23 And it shall come to pass, that every soul, which will not hear that prophet, shall be destroyed from among the people. 3:24 Yea, and all the prophets from Samuel and those that follow after, as many as have spoken, have likewise foretold of these days. 3:25 Ye are the children of the prophets, and of the covenant which God made with our fathers, saying unto Abraham, And in thy seed shall all the kindreds of the earth be blessed. 3:26 Unto you first God, having raised up his Son Jesus, sent him to bless you, in turning away every one of you from his iniquities.
That is NOT Paul's gospel.
1 Thessalonians 1:5 For our gospel came not unto you in word only, but also in power, and in the Holy Ghost, and in much assurance; as ye know what manner of men we were among you for your sake. 1:6 And ye became followers of us, and of the Lord, having received the word in much affliction, with joy of the Holy Ghost: 1:7 So that ye were ensamples to all that believe in Macedonia and Achaia. 1:8 For from you sounded out the word of the Lord not only in Macedonia and Achaia, but also in every place your faith to God-ward is spread abroad; so that we need not to speak any thing. 1:9 For they themselves shew of us what manner of entering in we had unto you, and how ye turned to God from idols to serve the living and true God;
Verse 1:10 And to wait for his Son from heaven, whom he raised from the dead, even Jesus, which delivered us from the wrath to come.