No argument from me on the theological point. But what causes Israel to be politically "illegitimate"? As I said, conquest has always been a legitimate means of creating or expanding a state, historically speaking. Although Israel was created by the UN, they fought to make it a reality, and then conquered more lands to expand it. Not pretty, but not "illegitimate" either.
They were given land which was not the giver's to give. When you are a custodian of someone else's property and you unlawfully give to a third party, that is nothing less than theft. And besides the modern Jews are not a "people", they are a mongrelized alien race which has little connection to the ancient Hebrews and even less of a right to claim the Holy Land as their inheritance.
In short, in temporal terms, legitimate is synonomous with lawfulness, and in the end what is and is not lawful issues from the natural law.
Pius X would not recognize that entity because it was not legitimate, and its creation was a violation of God's law and decrees. These things matter here among Christians, I would assume.
You base your claim upon the UN recognizing them, but that is ludicrous because the UN is illigitimate in and of itself, and was created largly according to Jewish principles and under the influence and power of Jews and Masonry, then and now.