I applaud your depth of study, and your desire to seek truth! We should all be at least as studious as you are! I beg you please, pray for guidance by holy spirit before you meditate on the below facts…
I understand where you are coming from on the ideas of the Israelites “scattered about” James 1:1 mentions, and obviously you consider Jesus’ words at Matt. 5 to be suggesting the same idea; so really the question is “Are there any scriptures to prove that the “…children of God…” that Caiaphas prophesied about at John 11:49-52 are ONLY Jews?” While I believe @SongofHannah gave you plenty, you did not accept any of those, so I submit to you 1 concept and two more scriptural references that should be irrefutable, IF you let the spirit lead you. (The 3rd and 4th references are your own, just read further.)
Since the formation of the Israelite nation and the exodus from Egypt there have been two kinds of Jews, Fleshly and Spiritual. The “Fleshly” Jews were born into it, you were a Jew until you died; in other words it is not possible for a Jew to become a Gentile. The Spiritual Jews (Gentiles who became proselytes and were called “alien-residents”) were those who chose to live as a Fleshly Jew, but didn’t have the blood, didn’t have a “tribe”, and also by Law from Jehovah, didn’t have a land-inheritance (but later Jehovah changed this). The entire tribe of Levi was set aside for Priesthood, out of the 12 tribes. The rest of the nation was to be holy, but could not be priests. Fast forward to after Jesus says “…your house is abandoned to you!…” and he says to the “little flock” (all faithful Jews) ‘I am going to bring in ones from outside to fill out your numbers, and they will all be under Me as the shepherd…’ Spiritual Israel was being formed, and it would need the same two parts, the priesthood (144,000 first-fruits of wheat harvest) and the holy nation (great crowd-all of the wheat harvest) it is bought out of. Jesus made this clear when He made with His disciples the Covenant for a Kingdom on His final night; He said ‘…you will rule over the 12 tribes of Israel…’
I am glad you referenced the 3rd Key given to Peter because the way he uses that Key is the first scriptural reference. You stated that his use of the 3rd Key was to go to the Jews scattered among the nations, but I would ask you to re-read Acts 10 and see if you still believe that. I know you are enough of a Beroean to know that Cornelius was a Roman, NOT a Jew, and of course his entire family would be Roman as well, NOT displaced Jews. The fact that there is NO SPECIFICATION of who will and won’t, or who can or can’t go to heaven is proven by Paul not being sure if he was going there, his stating that people should not partake wrongfully, his references to some of those congregations of their doing things for “the Holy Ones”, and the explanation he gives at Eph. 3:5-6 which makes an extremely clear reference to the whole point of the “sacred secret”, that being ‘the TRUTH of the makeup of Spiritual Israel’. Notice here Paul specifically used the term “joint heirs” in reference to those of the nations; if he were referring to natural Jews, lost or otherwise, they would always just be “heirs”, rightfully by blood, never needing to be called “joint heirs”.
I am glad you referenced Romans 9:3-5, though if you would have just read through vs. 8 you would have completed the picture; Paul here answers the question raised by what Caiaphas said! Ro. 9:8-That is, the children in the flesh are NOT really the children of God, but the children by the promise are counted as the seed. (Caps mine) This promise of course refers all the way back to Genesis and “The Seed”, and the cross reference also points to Isa. 54:1 which mentions the ‘more numerous’ ones born to the barren woman, those ones of course would be all of us that are NOT natural Jews.
Finally I would admonish you to go back to another scripture you referenced, Romans 11:1, and I would entreat you to please read the rest of the chapter. Using the illustration of the Olive tree and grafting, Paul makes very clear the difference with natural olive branches and wild olive branches, and this reference can’t be talking about Local Faithful Jews vs. Scattered Non-Faithful Jews because he specifically compares the natural ones that were cut off (scattered) and their grafting being DIFFERENT than the grafting of wild ones (gentiles).
I pray you have peace and joy in your further studies with your newfound depth of understanding! Keep digging for hid treasures!