Why could Paul quote Isa. 54:1 in Gal. 4:27

I don’t think that I was very clear on why Paul could connect Is. 54:1 with the allegory of Hagar and Sarah, Gal. 4:21-31. (Quoted in 4:27)
In the allegory, Paul is answering the false teachers, those who would pervert the gospel of free grace, by saying that to be truly a part of the children of God you had to be joined to the Israel of the Mosaic Law. It would appear that these Judaizers were claiming the authority of Jerusalem from Gal. 2:12 “For prior to the coming of certain men from James (from the Jerusalem that now is, earthly Jerusalem), he (Peter) used to eat with the Gentiles…” In Galatians Paul is identifying where spiritual authority rests; and those who are the true people of God.
What in the world does this verse in Isaiah have to do with the argument of where spiritual authority rests and who are the true people of God?
In Isa. 54 God is addressing the nation of Israel, which is pictured as a woman, first forsaken but then restored: Read Isa. 54:5-7
“For your husband is your Maker, Whose name is the LORD of hosts; And your Redeemer is the LORD of hosts; And your Redeemer is the Holy One of Israel, Who is called the God of all the earth.
For the LORD has called you, Like a wife forsaken and grieved in spirit, Even like a wife of one’s youth when she is rejected, Says your God.
For a brief moment I forsook you, But with great compassion I will gather you.”
The simple answer is that Israel, before the Babylonian exile, would picture Hagar. That is, Israel in covenant with God (her husband) producing children, the Israelites under the Old Covenant. While Israel, in exile, pictures a wife with no husband. Israel forsaken for a while by God (barren Sarah), v.7 forsaken “for a brief moment.” Isaiah speaks of the hope that there is a coming time when God would once again restore Israel, a time of rejoicing and shouting for joy. Isaiah 54:1 “For the sons of the desolate one will be more numerous than the sons of the married woman, says the LORD.”
Remember that the prophets spoke in two directions, especially beginning with Isaiah. 1. They were addressing the physical nation of Israel under the Old Covenant, calling the nation back to obedience to that covenant; however, 2. The prophets were also speaking of a day when a glorious New Covenant - God’s ultimate purpose in fulfilling the Abrahamic promises through Jesus Christ (Justification by Grace through Faith) – would produce more numerous sons.
Sarah is mentioned by name in the OT, outside the book of Genesis, only in
Isa. 51:1-2. There, it is said of her, that she has given birth to those who “pursue righteousness” and who “seek God”. It has been said by some that in the whole book of Isaiah, the barrenness of Sarah transforms from a story of a child to the future story of a people. ( the True People of God)
It is this looking forward to a people – to a people who will seek and pursue righteousness - The righteousness that is provided by the Cross work of Jesus Christ - that form the True Israel ( the Church ) of which Old Covenant Israel was only a temporary, unbelieving, picture.
Would this True Israel simply be an extension of the physical nation of Israel which Paul has already connected to Mt. Sinai, Hagar, and the Old Covenant by means of obedience to the Mosaic Law? Absolutely not! The true people of God would come from something new…the New Covenant which is NOT LIKE the one God established with Israel when He took them by the hand and led them out of Egypt to Himself at Mt. Sinai. (Jeremiah. 31:31)
This is the work of God Ezekiel 36 and 37 speak of, (i.e. the Valley of Dry Bones) pointing to the return of Israel from captivity but more importantly pointing to a greater restoration not just physical but spiritual, not just of one nation Israel, but of nations – people from every nation, making up the True people of God, the Church. (Jeremiah 31:31; Heb. 8:7-13)
It would not be the Law which would produce freedom for Israel; it would be God’s Grace. It will not be the Law that will produce the True people of God - Hagar can not produce free sons, only slaves. Gal. 4:24
If this is what Paul had in mind, then it becomes clearer why and how he could use this verse from Isaiah.
Isa. 54:1 follows immediately after Isa. 53 the Suffering Servant Song. And the context of Isa. 53:12 to Isa. 54:3 speaks of Inheritance.
Isa. 53:12 “Therefore, I will allot Him a portion with the great, And He will divide the booty with the strong; Because He poured out Himself to death, And was numbered with the transgressors; Yet He Himself bore the sin of many, And interceded for transgressors.”
Isa. 54:3 “For you will spread abroad the right and the left. And your descendants (seed) will possess nations And will resettle the desolate cities.”
There would be a restoration of physical Israel back into the land. But this restoration is only a picture of the Great Restoration that the Suffering Servant will bring about in producing, through a Work of the Holy Spirit, a people of EVERY tongue, nation, kindred, and tribe by faith in the Work of the Suffering Servant of Isa. 53, Jesus Christ dying for “transgressors” on the Cross of Calvary
Sarah’s children include those who truly search and seek for God (as revealed in the Person and work of Christ) and His righteousness. Righteousness “IN CHRIST”.
If we have been made apart of “the Jerusalem that IS free; she IS our mother.” The Jerusalem from above. WHY would anyone be so foolish to want to return to a slavish Jerusalem that had been abandoned by God?
The Judaizer’s claimed spiritual authority from Jerusalem, Paul is simply saying… you have the wrong Jerusalem! Gal. 4:25 “Now this Hagar is Mount Sinai in Arabia and corresponds to the PRESENT Jerusalem, for she (present Jerusalem) is in slavery with her children. Present Jerusalem has no authority.
True Authority and the True People of God are birthed from above by a free woman. Grace not Law! Gal. 4:26 “But the Jerusalem above is free, she is our mother”
“OUR” in v. 26 speaks of the true sons of Sarah, Everyone, Jew or Gentile, who has come by the Sovereign work of the Spirit of God by Grace through Faith to rest and trust only in Jesus Christ for their acceptance before God.
WHO? after coming to know and experience the Freedom of the Gospel, would ever return to the slavery of the Law? And verse 30 truly sets forth that the two, Grace and Law can not be mixed. “Cast out the bondwoman and her son, For the son of the bondwoman shall not be an heir with the son of the free woman”
v. 31 “So then, brethren, WE are not children of a bondwoman, but of the free woman.”
REJOICE and live as free sons. Pastor Mike