I agree with you on Ezekiel 35, but according to Scripture Edom still exists. The Scriptures clearly show that Edom assumed true Israel's identity when Israel was forced out of Samaria and Judah. The so-called Jew today is Edom according to Scripture. Please read Jeremiah 25. Moreover, read Ezekiel 35:9-15 again. According to the word, Edom still has Samaria and Judah in its possession. I will also suggest that you read, Israel's Prime Minister Begin's Chicago Tribune Sept 9, 1982 article that clearly confirms The Lord's word in Ezekiel 35:10, Mr. Begin boastfully proclaims, "that Samaria and Judah would be theirs forever."
Moreover, it was Edom's old homeland (Seir) that was laid to waste, but Edom or Idumea has assumed Israel's homeland thereby assuming its identity. Furthermore, the word clearly states that there would be a perpetual hatred between Esau (Edom) and Jacob ( Israel) even to this day. According to scripture true Israel was and is still scattered and will not be returned to Jerusalem until the return of Christ.
Though you claim "according to the Scriptures," your words are empty as you did not show any scriptural support for your claims. In Jeremiah 25, Edom is mentioned once in a large list of nations who would be destroyed along with Judah (Jeremiah 25:17-29). You claim Edom took over Israel's identity, but Jeremiah doesn't say that. And why would you pick one nation out of a large list? Why not the Egyptians, Philistines, Moabites, Ammonites, Phoenicians, etc?
Ezekiel 35 discusses the destruction of Edom in more detail, "Thus I will make Mount Seir most desolate, and cut off from it the one who leaves and the one who returns. And I will fill its mountains with the slain; on your hills and in your valleys and in all your ravines those who are slain by the sword shall fall. I will make you perpetually desolate, and your cities shall be uninhabited; then you shall know that I am the LORD" (Ezekiel 35:7-9). The reason Edom would be destroyed is that they thought they could possess their land and Israel's. "Because you have said, 'These two nations and these two countries shall be mine, and we will possess them,' although the LORD was there, therefore, as I live," says the Lord GOD, "I will do according to your anger and according to the envy which you showed in your hatred against them; and I will make Myself known among them when I judge you" (Ezekiel 35:10-11). This is not talking about current events, it is talking about what Edom thought it could do during the Babylonian captivity of Israel.
As Jeremiah points out, it wasn't just Edom. "They have said, "Come, and let us cut them off from being a nation, that the name of Israel may be remembered no more." For they have consulted together with one consent; they form a confederacy against You: the tents of Edom and the Ishmaelites; Moab and the Hagrites; Gebal, Ammon, and Amalek; Philistia with the inhabitants of Tyre; Assyria also has joined with them; they have helped the children of Lot" (Psalm 83:4-8). God promised to deal with them harshly in return. "Therefore, O mountains of Israel, hear the word of the Lord GOD! Thus says the Lord GOD to the mountains, the hills, the rivers, the valleys, the desolate wastes, and the cities that have been forsaken, which became plunder and mockery to the rest of the nations all around- therefore thus says the Lord GOD: "Surely I have spoken in My burning jealousy against the rest of the nations and against all Edom, who gave My land to themselves as a possession, with whole-hearted joy and spiteful minds, in order to plunder its open country." Therefore prophesy concerning the land of Israel, and say to the mountains, the hills, the rivers, and the valleys, thus says the Lord GOD: "Behold, I have spoken in My jealousy and My fury, because you have borne the shame of the nations." Therefore thus says the Lord GOD: 'I have raised My hand in an oath that surely the nations that are around you shall bear their own shame'" (Ezekiel 36:4-7).
Where nations, such as Edom would bare shame permanently because they would rise no more, Israel would return. "But you, O mountains of Israel, you shall shoot forth your branches and yield your fruit to My people Israel, for they are about to come. For indeed I am for you, and I will turn to you, and you shall be tilled and sown. I will multiply men upon you, all the house of Israel, all of it; and the cities shall be inhabited and the ruins rebuilt" (Ezekiel 36:8-10).
So yes, Edom, along with a number of other nations thought they could kick Israel when it was down, possess its territory, and plunder its goods, but God said they would be punished in turn and that Israel would regain its land. The Scriptures do not say that Edom assumed the identity of Israel.