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Are you unaware of Pius IX's discussion of "invincible ignorance?"

https://www.papalencyclicals.net/pius09/p9quanto.htm

"7. Here, too, our beloved sons and venerable brothers, it is again necessary to mention and censure a very grave error entrapping some Catholics who believe that it is possible to arrive at eternal salvation although living in error and alienated from the true faith and Catholic unity. Such belief is certainly opposed to Catholic teaching. There are, of course, those who are struggling with invincible ignorance about our most holy religion. Sincerely observing the natural law and its precepts inscribed by God on all hearts and ready to obey God, they live honest lives and are able to attain eternal life by the efficacious virtue of divine light and grace. Because God knows, searches and clearly understands the minds, hearts, thoughts, and nature of all, his supreme kindness and clemency do not permit anyone at all who is not guilty of deliberate sin to suffer eternal punishments."

If Pius IX thought it important enough to mention, why wouldn't it be important for the trad clergy to make that distinction as well?

The distinction Pius IX is making is between those who never held the Catholic Faith (because they were ignorant of it) and those who did, at one time, hold the Catholic Faith but fell away from the Faith and "are living in error."

Pius IX did not make the distinction in the same way clergy do today. In fact, I don't think the "distinction" Pius IX made is what you are claiming it is at all.

very grave error entrapping some Catholics who believe that it is possible to arrive at eternal salvation although living in error and alienated from the true faith and Catholic unity. Such belief is certainly opposed to Catholic teaching. There are, of course, those who are struggling with invincible ignorance about our most holy religion

Notice that, after saying that the belief that those "living in error and alienated from the true faith and Catholic unity" can "arrive at eternal salvation" is "certainly opposed to Catholic teaching", Pius IX does NOT say "however" or "but" or any other common beginning of EENS soft denial.  He teaches the dogma as it is, and leaves it at that, and then moves on to the invincibly ignorant.

And what does he say about the invincibly ignorant? Not what you claim, and not what trad clergy teach. He simply says "they live honest lives and are able to attain eternal life by the efficacious virtue of divine light and grace". Where in this quote do we read that the invincibly ignorant can be saved without formally entering the Church? Without explicit faith and baptism?
Those invincibly ignorant who live honest lives and observe the natural law will be furnished with the means of salvation. That must happen, in order for them to be saved, because the invincibly ignorant are included in those who "[live] in error and [are] alienated from the true faith and Catholic unity"! And what did Pius IX teach about those "living in error and alienated from the true faith and Catholic unity."? They cannot arrive at eternal salvation


Was it Pius IX that lamented liberals who twisted and misinterpreted his words?

Pope Eugene's comment also references those who do not "remain" in the Catholic Church, meaning that they were once Catholic and fell away from the Faith. See the word "remain" in your quote from Florence:

"The most Holy Roman Church firmly believes, professes and preaches that none of those existing outside the Catholic Church, not only pagans, but also Jews and heretics and schismatics, can have a share in life eternal; but that they will go into the 'eternal fire which was prepared for the devil and his angels' (Matthew 25:41), unless before death they are joined with Her; and that so important is the unity of this ecclesiastical body that only those remaining within this unity can profit by the sacraments of the Church unto salvation, and they alone can receive an eternal recompense for their fasts, their almsgivings, their other works of Christian piety and the duties of a Christian soldier. No one, let his almsgiving be as great as it may, no one, even if he pour out his blood for the Name of Christ, can be saved, unless he remain within the bosom and the unity of the Catholic Church."

-Council of Florence

Pope Eugene would not say "pagans", "Jews and heretics and schismatics" if he was speaking of Catholics, because none of those people are Catholic. Again, note that he made no distinction between pagans, Jews, heretics, and schismatics who are invincibly ignorant and those who are not. They will all go into the eternal fire unless, before death, they are joined with the Church. If they are joined with the Church before death, they are no longer pagans, Jews, etc.

The usage of the words "remaining" and "remain" appear AFTER Pope Eugene says that those listed above must be joined with the Church in order to be saved. You must be joined with the Church AND remain in Her unity in order to be saved

Bottom line: according to Pius IX, "those who are struggling with invincible ignorance about our most holy religion" AND "observing the natural law and its precepts inscribed by God on all hearts and ready to obey God," even though they are outside of the Catholic Church, will not be eternally punished.

Do you agree or disagree? If you agree, what do you think Pius IX means? If they will not be eternally punished, what will happen to them when they die?

I disagree with your interpretation of what Pius IX was saying. I believe those who fall under the conditions laid out by Pius IX will be provided with the necessary means of salvation, and if they accept those means they will be saved. When you say they are "outside of the Catholic Church" yet "will not be eternally punished" you are saying something Pius IX never taught, and you should not imply that he ever taught that because that is contrary to Church teaching. Pius IX said the invincibly ignorant are "able to attain eternal life by the efficacious virtue of divine light and grace.". As to the hypothetical invincibly ignorant person who observes the natural law and is ready to obey God, yet never received the opportunity to enter the Church by Baptism, there is no evidence for such a person ever having existed 


Do you agree or disagree that ALL non-Catholics are "living in error and alienated from the true faith and Catholic unity"?

At the end of the day, it is human cowardice.  They do not want to have to tell Jimmy, the Hindu next door, that if he does not become a Catholic, then he will be damned. It is much easier to water things down.  These same priests take the same position with the New Mass. Oh, we cannot tell the people that the New Mass is a sin, we do not want to hurt their FEEWINGS!!!!

THIS ^^^

Has any soul ever been lost because someone told them there's no salvation outside the Catholic Church?  No.  In fact, many have been saved.

But many are lost due to a complacency that comes from being told the opposite.

Even among Trad clergy many have weak minds and weak faith, despite having some sensibilities in favor of smells and bells.

They are also intellectually dishonest ... since they hold th exact same ecclesiology that attack V2 over as its core heresy.