It would seem that you are simply unable to read JP II's plain words:
9. Jesus himself, at the hour of his Passion, prayed "that they may all be one" (Jn 17:21). This unity, which the Lord has bestowed on his Church and in which he wishes to embrace all people, is not something added on, but stands at the very heart of Christ's mission.
That is exactly what Leo XIII, Pius XI, and Pius XII, said.
Now here is the "false opinion":
For authors who favor this view are accustomed, times almost without number, to bring forward these words of Christ: "That they all may be one.... And there shall be one fold and one shepherd,"[14] with this signification however: that Christ Jesus merely expressed a desire and prayer, which still lacks its fulfillment. For they are of the opinion that the unity of faith and government, which is a note of the one true Church of Christ, has hardly up to the present time existed, and does not to-day exist.
It is that "the unity of faith and government ... does not to-day exist", and that this may be deduced from "Ut unum sint". Compare to JP II's teaching in UUS, which is the exact opposite of the false opinion:
11. The Catholic Church thus affirms that during the two thousand years of her history she has been preserved in unity, with all the means with which God wishes to endow his Church, and this despite the often grave crises which have shaken her, the infidelity of some of her ministers, and the faults into which her members daily fall.
Also see his comments in the CCC, which note the same prayer, "ut unum sint":
815 What are these bonds of unity? Above all, charity "binds everything together in perfect harmony."265 But the unity of the pilgrim Church is also assured by visible bonds of communion:
- profession of one faith received from the Apostles;
- common celebration of divine worship, especially of the sacraments;
- apostolic succession through the sacrament of Holy Orders, maintaining the fraternal concord of God's family.266820 "Christ bestowed unity on his Church from the beginning. This unity, we believe, subsists in the Catholic Church as something she can never lose, and we hope that it will continue to increase until the end of time."277 Christ always gives his Church the gift of unity, but the Church must always pray and work to maintain, reinforce, and perfect the unity that Christ wills for her. This is why Jesus himself prayed at the hour of his Passion, and does not cease praying to his Father, for the unity of his disciples: "That they may all be one. As you, Father, are in me and I am in you, may they also be one in us, . . . so that the world may know that you have sent me."278 The desire to recover the unity of all Christians is a gift of Christ and a call of the Holy Spirit.279
You're cherry-picking again. See post 85. I've answered this already. Pius IX, Leo XIII, Pius X, XI and XII never had the contradictory statements or the mealy mouthed gobbledygook that JPII puts in his ridiculous encyclicals.