Q. How can anyone really believe that the pope was infallibly informed about the immaculate conception in 1854 and Assumption of Mary in 1950, so late in the history of the Church?
A. I would have trouble believing these doctrines too if they were first introduced nearly two centuries after Christ. However, that is not what the dates mentioned mean. The Christian Faith was taught by the apostles and those they ordained to continue the ministry. The Pope and bishops have never sat down and solemnly and systematically declared all of the doctrines of the Catholic Faith. That isn’t how she works. She evangelizes, baptizes, teaches, and administers the sacraments. Doctrines are not defined until and unless they become seriously threatened by heresy.
For instance, controversy arose in ACTS about whether Gentiles had to keep all of the Jewish laws and be circumcised. The controversy was examined and settled at the very first Church Council in Jerusalem (Acts 15). Most of the heresies in the early Church involved wrong ideas about Jesus. Was he merely a holy man? Was he a god that appeared to be a man? Was he a man who became a god at his baptism? Was he fully God and fully man? All of these were settled at the Council of Nicea in 325 AD when the Doctrine of the Trinity was defined. This doctrine was not invented or communicated to the pope for the first time at this date. The doctrine existed from the very beginning but when heresy grew and threatened the truth, the Pope and the bishops met in council and settled the issue clearly and once and for all.
Again, later, due to confusion and controversy the Pope and Council of Hippo defined the contents of the New Testament in 393 AD and 397 AD. The pope did not invent the Doctrine of the Trinity in 325 AD or the New Testament in 397 AD. This is exactly the case with the Doctrines of the Immaculate Conception and Assumption. They have been believed from the beginning. But with the rise of Protestantism these doctrines became more and more threatened and so were solemnly and clearly defined.