I agree that verses 22-27 are words spoken by Gabriel.
23 At the beginning of thy supplications the commandment came forth, and I am come to shew thee; for thou art greatly beloved: therefore understand the matter, and consider the vision.
understand the matter - is that of the 70 years of serving the King of Babylon being about over.
consider the vision - is to consider the vision which Daniel first encountered Gabriel. Which was of the time of the end vision about the little horn person to stop the daily sacrifice and commit the transgression of desolation .
No, Daniel did not call his people the "Israel of God"
That phase is found in one place in the bible - Galatians 6:16.
No, "thy people" i.e. Daniel people, are the ethnic Jews.
In Daniel 10, Daniel received another visit from an angel who told him everything that was going to happen to his people in Daniel 11-12. With Daniel 11:36-12:13 being of things in the latter days, end times.
Daniel 10:14 Now I am come to make thee understand what shall befall thy people in the latter days: for yet the vision is for many days.
Daniel 8 > Daniel 9 > Daniel 10 (introducing Daniel 11 > Daniel 12)
That's the whole big picture of what is going to happen to Daniel's people, the ethnic Jews.
The prophecy still pertains to the people of Daniel's faith, not ethnicity. And both chapters speak of what shall come to pass in the latter or last days. The latter days or "these last days." I don't find a single prophet prophesying of what shall come to pass in the last/latter days limiting "thy people" or people of Daniel's ethnicity only to ethnic Jews. Rather the prophets foretell the prophesy for the last/latter days shall be to ALL people throughout the earth who are of faith.
Isaiah 2:2 (KJV) And it shall come to pass in the last days, that the mountain of the LORD'S house shall be established in the top of the mountains, and shall be exalted above the hills; and all nations shall flow unto it.
Micah 4:1-2 (KJV) But in the last days it shall come to pass, that the mountain of the house of the LORD shall be established in the top of the mountains, and it shall be exalted above the hills; and people shall flow unto it. And many nations shall come, and say, Come, and let us go up to the mountain of the LORD, and to the house of the God of Jacob; and he will teach us of his ways, and we will walk in his paths: for the law shall go forth of Zion, and the word of the LORD from Jerusalem.
Acts 2:17 (KJV) And it shall come to pass in the last days, saith God, I will pour out of my Spirit upon all flesh: and your sons and your daughters shall prophesy, and your young men shall see visions, and your old men shall dream dreams:
Hebrews 1:1-14 (KJV) God, who at sundry times and in divers manners spake in time past unto the fathers by the prophets, Hath in these last days spoken unto us by his Son, whom he hath appointed heir of all things, by whom also he made the worlds; Who being the brightness of his glory, and the express image of his person, and upholding all things by the word of his power, when he had by himself purged our sins, sat down on the right hand of the Majesty on high; Being made so much better than the angels, as he hath by inheritance obtained a more excellent name than they. For unto which of the angels said he at any time, Thou art my Son, this day have I begotten thee? And again, I will be to him a Father, and he shall be to me a Son? And again, when he bringeth in the firstbegotten into the world, he saith, And let all the angels of God worship him. And of the angels he saith, Who maketh his angels spirits, and his ministers a flame of fire. But unto the Son he saith, Thy throne, O God, is for ever and ever: a sceptre of righteousness is the sceptre of thy kingdom. Thou hast loved righteousness, and hated iniquity; therefore God, even thy God, hath anointed thee with the oil of gladness above thy fellows. And, Thou, Lord, in the beginning hast laid the foundation of the earth; and the heavens are the works of thine hands: They shall perish; but thou remainest; and they all shall wax old as doth a garment; And as a vesture shalt thou fold them up, and they shall be changed: but thou art the same, and thy years shall not fail. But to which of the angels said he at any time, Sit on my right hand, until I make thine enemies thy footstool? Are they not all ministering spirits, sent forth to minister for them who shall be heirs of salvation?