The video assumes Paul's definition of "Israel". That is, the video explicates what Paul meant to say in Romans 11. Paul intends to say something about Israel as he and his readers understood it.
Let's review the text.
I say then, they did not stumble so as to fall, did they? May it never be! But by their transgression salvation has come to the Gentiles, to make them jealous. Now if their transgression is riches for the world and their failure is riches for the Gentiles, how much more will their fulfillment be! But I am speaking to you who are Gentiles. Inasmuch then as I am an apostle of Gentiles, I magnify my ministry, if somehow I might move to jealousy my fellow countrymen and save some of them. For if their rejection is the reconciliation of the world, what will their acceptance be but life from the dead? If the first piece of dough is holy, the lump is also; and if the root is holy, the branches are too.
Paul rhetorically asks whether Israel fell when she stumbled, when did she stumble? Elsewhere we learn that She tripped over the cornerstone that God laid in Zion. We know that the cornerstone is Jesus, the Christ, the King of Israel. Putting Jesus on a cross was her blunder, her misstep, which Paul says turned out as riches for the Gentiles.
By implication then, since Paul is contrasting Israel with the Gentiles, we know that Israel does not include Gentiles. Israel is being contrasted with Gentiles, and in this context it is understood that the contrast is between the nation of Israel and the other nations in the world.
I'm surprised at how few actually understand all those of Romans 11 that Paul was pointing to. Paul's brethren the Jews are only about 3 tribes of Israelites, not all the seed of Israel. I know why many brethren don't understand this; it's because many don't like the subject of history, so they don't study their Old Testament history either, even though studying Bible history is essential to understanding much of The New Testament Scriptures, because those like the Apostles understood Old Testament history. That is even why Apostle Peter admonished those in Christ to be mindful of the words of the prophets too (2 Peter 3:2).
1. God split old Israel into 2 separate kingdoms because of Solomon allowing his many foreigner concubines and wives to bring idol worship into Israel. (1 Kings 11).
2. The southern kingdom under Solomon's son, Rehoboam, was made up of only 3 tribes, the tribes of Judah, Benjamin, and Levi. These became known as JEWS, that name derived from the tribe of Judah. This kingdom with its capital city at Jerusalem, became known per God's Word as the "kingdom of Judah", or "house of Judah", or just "Jerusalem", or "Judah", after the split. (see 1 Kings 12 to 2 Kings 17).
3. The northern kingdom that God created at that split, was the "kingdom of Israel", and was made up of the ten northern tribes of Israel that lived in the northern lands. God made Jeroboam, born of the tribe of Ephraim, king over Israel, with its capital city at Samaria in the north. This northern kingdom was known as the "house of Israel", "Ephraim", "Joseph", "Israel", or "Samaria", all labels used to point to this northern ten tribe kingdom after the split of 1 Kings 11.
4. God gave prophecies specific to each 'house'. For example, the Book of Hosea is mainly a Book of prophecy God gave about the ten northern tribe "kingdom of Israel", or "house of Israel". There is very little mention of Judah and the southern kingdom there. Likewise, most of the Book of Ezekiel is written to the ten tribe "house of Israel", because God told Ezekiel He made him a prophet to the "house of Israel". And most of the Books of Jeremiah and Daniel are written about and to the Jews. These separate prophecies in God's Word have continued in history to this day. So if one doesn't know this history, then they cannot properly understand just 'who' those prophecies are about, and who they are for, and won't even recognize them when they come to pass.
5. Jeroboam, king of Israel of the ten tribe northern "kingdom of Israel" feared that the ten tribes would travel south to Jerusalem to worship (proper worship then). So Jeroboam made two golden calf idols for the ten northern tribes to worship in the north, and he also made common priests of the people. Thus the Levites living among the northern ten tribes couldn't perform their duties, so they left and went south and joined with the tribes of Judah and Benjamin. A small remnant out of the ten northern tribes that refused Jeroboam's idols also left and went south to join with Judah. The majority of Israelites stayed in the north with Jeroboam, king of Israel (2 Chronicles 11).
6. The two kingdoms had war against each other.
7. After a while, God finally got fed up with Jeroboam's calf idol worship and rebellion by the ten northern tribes. Per 2 Kings 17, in the days of king Hoshea of the northern kingdom, God caused the kings of Assyria to remove the ten tribes out of the holy land, and made them captive to Assyria and the cities of the Medes. Only the southern kingdom of JEWS of the "house of Judah" was left remaining in the holy land at Jerusalem-Judea.
8. Once the kings of Assyria removed the ten northern tribes of Israel, they placed five pagan peoples from Babylon in the northern holy lands instead. These became known as the Samaritans. The Jews of the southern "kingdom of Judah" would not have anything to do with them, the Samaritans being foreign pagan Gentiles. Understanding this history of the five pagan peoples from Babylon links with what Jesus was speaking about with the Samaritan woman at the well.
9. God prophesied what He would do to the northern ten tribes of the "kingdom of Israel" once He removed them captive to Assyria and cities of the Medes. He gave many of these prophecies through His prophet Hosea. The ten tribes would be further scattered, and would go fully into their false Baal worship they desired, which caused them to forget Him, and they would lose knowledge of their heritage as the larger portion of the people of Israel. God would not lose them, but they, the Jews of the southern kingdom, and the world, would lose the identity of the ten tribes of Israel, as it is among the majority still today (but a remnant was always to understand per God's promise).
10. After the majority of Israelites of the northern kingdom were removed, that left only the 3 tribe "kingdom of Judah" in the southern land. These became known as Israel, because the 3 tribes of Judah, Benjamin, and Levi indeed are Israelites, though only a small portion of total Israelites. And they were the only seed of Israel left in the holy land after God removed the ten northern tribes (see 2 Kings 17:18). They and the strangers living with them, all became known as JEWS (per the Jewish historian Josephus). That title did not apply to the scattered ten northern tribes. It only applied to those of the southern "kingdom of Judah" that were under the reign of the house of David in Jerusalem.
11. About 120 years after God had scattered the ten northern tribe kingdom, the "house of Judah" also began to fall away from Him and rebel. So God brought Nebuchadnezzar, king of Babylon upon this 3 tribe kingdom of Judah in the south, and removed them captive to Babylon for 70 years. Neb destroyed the city of Jerusalem and Solomon's temple. After the 70 years, a small... remnant of Jews returned from Babylon back to Jerusalem, and rebuilt the city, wall and 2nd temple. Because God had taken care of the Jews while captive in Babylon, the majority of Jews of the "house of Judah" chose to stay in Babylon. But a majority of those Jews that stayed would be scattered later through the countries from Babylon, like the ten tribes had been scattered.
12. So when God's Word points to the Jews of the "house of Judah" after the split, you are not seeing the ten northern lost tribes of Israel there at all, which actually made up the majority of Israelites of old Israel. And even then, after the Jews return from their Babylon captivity, you are only seeing a small remnant of the Jews, the larger portion having chosen to remain in Babylon.
Many of today's Jews like to claim they only are God's chosen people, but that is not Biblically nor historically true about the chosen seed of Israel. Apostle Paul quoted from the Book of Hosea about the "house of Israel" linked with ALL believers on Jesus Christ in final (see Romans 9:24-26, a link to Hosea prophecy). And before coming to Romans 11, one is supposed... to understand why Paul linked the ten tribe "house of Israel" with believing Gentiles per the prophecies in the Book of Hosea that he quoted.