Greetings in that Divine and Most Precious Name of Our Lord and Savior Jesus Christ!
It is no surprise that you are coming to a Latin conclusion because like the Latins, you are thinking way to scholastically, not spiritually enough. The Church is filled with Mysteries, which only God can explain. The concept of Infallibility is a bit comical if you ask me, it tries to make the Holy Spirit a matter of reason, logic, mathematical precision. That is silly and borderline anthropomorphic..
The Bible is not infallible neither are the priests nor the Councils in the legalistic or scholastic sense, which is to say, that word for word, letter for letter, punctuation mark for punctuation mark the perfection of God. The economy of the Holy Spirit is not so simple, it is a process, an event, a becoming, a verb. That means that the Bible or the Councils become infallible in prayer, through a direct communion of the Holy Spirit. The words, letters, and pages themselves are realistically no different than any other book, which is why they can be damaged, burned, scoffed, or disregarded. If the words and writings themselves were infallible, they would also be indestructible. They are paper, like we are flesh and blood. It is the Holy Spirit behind and underlying the words which is infallible. We connect with the Holy Spirit when we read and meditate and pray on the words, but it is the Holy Spirit that speaks to us, not necessarily the words themselves. The doctrine of Scriptural Infallibility when taken in a fundamentalist perspective, assumes that the words are perfect, and we are not, so we readjust our lives to the perfect words. The reality is the words were written, recorded, and transmitted by imperfect human hands, and so they rightfully and naturally reflect this same human imperfection. The Economy of God is to act in synergy with our imperfections through His own inherent perfection. So, again, God is perfect (infallible) and everything else is in a process of becoming infallible by Grace (i.e. synergy). We get closer to God when we read and pray on the Bible, but the Bible is not perfect aside from God. We connect to the perfect God when we pray and meditate on the Scriptures.