Was Israel the Jewish people's to begin-with?

Israel and Judah (c. 930 – 576 BCE)

Around 930 BCE, the kingdom split into a southern Kingdom of Judah and a northern Kingdom of Israel.

An alliance between Ahab of Israel and Ben Hadad II of Damascus managed to repulse the incursions of the Assyrians, with a victory at the Battle of Qarqar (854 BCE). However, the Kingdom of Israel was eventually destroyed by Assyrian king Tiglath-Pileser III
around 750 BCE. The Philistine kingdom was also destroyed. The
Assyrians sent most of the northern Israelite kingdom into exile, thus
creating the "Lost Tribes of Israel". The Samaritans
claim to be descended from survivors of the Assyrian conquest. An
Israelite revolt (724–722 BCE) was crushed after the siege and capture
of Samaria by Sargon II.

Assyrian King, Sennacherib, tried and failed to conquer Judah. Assyrian records claim he punished Judah and then left (Herodotus also described the invasion).

Early Israelites (1200–930 BCE)

The Merneptah Stele (JE 31408), the earliest record of the name "Israel" (Cairo Museum)

The first record of the name Israel (as ysrỉꜣr) occurs in the Merneptah stele, erected for Egyptian Pharaoh Merneptah c. 1209 BCE, "Israel is laid waste and his seed is not."^[10] William Dever
sees this "Israel" in the central highlands as a cultural and probably
political entity, but an ethnic group rather than an organized state.^[11]

Ancestors of the Israelites may have included Semites who occupied Canaan and the Sea Peoples.^[12]
McNutt says, "It is probably safe to assume that sometime during Iron
Age I a population began to identify itself as 'Israelite'",
differentiating itself from the Canaanites through such markers as the
prohibition of intermarriage, an emphasis on family history and
genealogy, and religion.^[13]

Villages had populations of up to 300 or 400,^[14][15] which lived by farming and herding and were largely self-sufficient;^[16] economic interchange was prevalent.^[17] Writing was known and available for recording, even in small sites.^[18]
The archaeological evidence indicates a society of village-like
centres, but with more limited resources and a small population.^[1