What did she appeal? You said she was never excommunicated.
"And as Bl. Pius IX says: 'For any man to be able to prove his Catholic faith and affirm that he is truly a Catholic, he must be able to convince the Apostolic See of this'"
Of course he also said, "If a future pope teaches anything contrary to the doctrines of the Church, do not follow him." Besides, it would be pretty pointless to try to convince an Apostolic See of your Catholicism when that See itself was abandoning the faith.
You are full of useless quotations. But in this case the citation is ironic, since it is the Pope who has need to prove his faith. It is he who opposed Catholic tradition and elevated modernists and heretics and did all he could to destroy the ancient Mass, not the Archbishop who had never doubted a smidgen of the Catholic faith or the tradition which conveyed it. So it is an irony of ironies to think anybody could appeal to John Paul. There could be no appeal--the fix was in. The Pope himself, in fact, ignored his own canon law to make sure the Archbishop was condemned--and then, by a leap of false logic, incorrectly equated a supposed disobedience with schism itself.
"Most men feel that the Church's supreme head and shepherd should decide who are Catholics and who are not" (Quartus Supra 8-9, 15)."
Jesus had a better yardstick. He knew some Church leaders would wear sheep's clothing but be ravening wolves underneath. So he warned us to look to their fruits. By this we would be able to discern the real shepherds from the false ones. A good tree bears good fruit. A bad tree bears rotten fruit. Nor can we look to these bad shepherds for guidance in the faith.