Why does Paul say he was born out of due time?

Answer Request:
Why does Paul literally call himself an ἐκτρώμα "abortion" at 1 Cor. 15:8?

1 Corinthians 15:7 Then He [Jesus] appeared to James, then to all the apostles. 8 And last of all He appeared to me also, as to one of untimely birth [prematurely as of one born out of due time].

Besides "abortion", as you mention, the translation of the Greek word also refers to,

• an untimely birth
• an immature birth

To clarify why Paul describes himself as one of immature birth, you need to understand this brief background of the purpose of Israel in God's redemption plan.

Exodus 19:4-6 You have seen what I did to the Egyptians, and how I bore you on eagles wings and brought you to Myself. 5 Now therefore, if you will indeed obey My voice and keep My covenant, then you shall be a special treasure to Me above all people; for all the earth is Mine. 6 And you shall be to Me a kingdom of priests and a holy nation. These are the words which you shall speak to the children of Israel.

The nation of Israel is called by God to be a kingdom of priests. Israel has not yet fulfilled this preeminent position due to unbelief and not recognizing and receiving their Messiah, Jesus Christ at His first advent. However, when Jesus Christ comes again at His second advent, Israel will receive Him and they will inherit the Millennial Kingdom. In this kingdom they will be preeminent and hold the position as priests over the nations, fulfilling a role as a mediator between God and the nations.

Zechariah 8:23 Thus says the Lord of hosts: 'In those days ten men from every language of the nations shall grasp the sleeve of a Jewish man, saying, "Let us go with you, for we have heard that God is with you.

Now, back to your question about Paul's immature birth.

This priestly position was part of a covenant God had made to Israel, but since Israel missed the opportunity at Christs first advent, God remained true to His word and use Paul as a 'proxy' Israel (or a premature Israel) to fulfil this position and bring His Word and blessings to the Gentiles.

Paul was "born out of due time" (prematurely) in the sense that he could serve in the priestly role as a proxy, as a representative for "believing" Israel, to propagate (transmit) the blessings (of the new covenant) through Israel to the Gentiles. (See Abraham's Blessing: Jeremiah 31:31-34; Galatians 3:14)

Romans 15:15-17 However, I have written you a bold reminder on some points, because of the grace God gave me 16 to be a minister of Christ Jesus to the Gentiles in the priestly service of the gospel of God, so that the Gentiles might become an acceptable offering to God, sanctified by the Holy Spirit. 17 Therefore I exult in Christ Jesus in my service to God.

Paul wrote that one day, "all Israel will be saved" (Romans 11:26). The "all Israel" will be Jews who will repent before the second advent of Christ at the end of the Tribulation. They will enter the Millennial kingdom and begin to serve as priests (Exodus 19:4-6, Zachariah 8:22-23) to the Gentile nations. Until that day comes, Paul served as a proxy, or as representing a redeemed, repentant Israel, in the role of priest. Since Paul was a Jew (Phil.3:5-6), since God commissioned him as the apostle to the Gentiles (Romans 11:13), God's divine plan to redeem "the body (church)", was faithful to his promise to Abraham (ie: in line with the covenant stipulations given to Abraham that through Israel all the families (Gentile nations) of the earth will be blessed). When Israel receives their Messiah, at the end of the tribulation, they will again fulfil this priestly position over the Gentile nations throughout the Millennial reign of Christ on earth.