A quote from you...”The House of Israel was scattered throughout all the nations, becoming Gentiles as the prophet Hosea said would happen.”
Yes, the House of Israel are not Jews. 1Kings 11, after Solomon died Reohboam his son, split the Kingdom in TWO. The House of Israel were the ten northern tribes and the House of Judah was Judah and Benjamin. Later God divorced the House of Israel but stayed in covenant with the House of Judah from which the Messiah would come. Jesus/Yahshua was a Jew from the tribe of Judah.
The Ten northern tribes were not JEWS!!!!! They were Hebrews, they were Israelites but they were not JEWS.
The ten northern tribes were later scattered into all the nations and became Gentiles. 2 Chronicles 5:26 Hosea 1-10.