Satis Cognitum, Leo XIII: “[...]”
If the Church lacks a legitimate hierarchy with the authority to carry out the mission Christ entrusted to His Church, it follows that the Church as Christ founded it no longer exists, which is contrary to the teaching of Pius XI.
Papal teaching is sure better than an encyclopedia. The original Latin of
Satis Cognitum can be found in the ACTA SANCTAE SEDIS on
vatican.va:
http://www.vatican.va/archive/ass/index_ge.htm (ASS 28, 1895-96)
http://www.vatican.va/archive/ass/docuмents/ASS-28-1895-96-ocr.pdf
At a first glance, the English translation is rather poor. E.g. the first "to the end of time" doesn't exist in the Latin.
It may take some time to reply to all your highlighted statements. You'll have to be patient.
For the time being, let me say that I do not expect that it will be possible to understand Leo XIII as contradicting
Pastor aeternus of the Vatican Council, saying that there will be shepherds "usque ad consummationem saeculi".