Ekklesia does not mean "CHURCH" as in the institution of a Church.
Ekklesia, the greek word translated most often into English as "(the) Church" just means assembly.
It doesn't refer to a man-made institution or organisation, it refers to the spiritual assembly of all believers, visa vie: where two or three gather in My name, there I am with them.
And that the gates of hell shall not prevail … well, yeah. Even if every Catholic priest, bishop and pope was deaded and every other apostolic equivalent, and every church building demolished, every block of land sold, every crucifix and every rosary burned, and so on … do you REALLY think the Church had been prevailed against? The Church is God's people, not some corporation chill, I'm using the term with its real meaning that claims "a direct line" from Peter and the other apostles exists in its bishops and other office holders. WE are the Church. Ergo, every last believer will have to be hunted down and killed before the Church has been prevailed against, and not even three generations of Kims has succeeded in that goal in one country.
And while we're on it: the line of apostolic succession, which, by magic, makes this guy who is Bishop of some place a direct successor to the original guy who held that office and was an apostle, ergo and ipso facto, that guy is an apostle by succession. Bishops, like everyone else, get elected. Granted it's by a select few (eg; the conclave), or maybe even only by one, their Archbishop, but it means this claim to "succession" isn't even a spiritual succession – as in the apostle Peter anointed Linus as HIS apostle, and Linus anointed Cletus, and so on. At least hereditary kings have some form of lineage. No, here it's the persistence of the post's existence, I guess. "We can trace a line of elected Bishops of Rome back to Christ's own apostle." Really? THIS is the entirety of the great claim to being "Princes of the Universe"?
It IS true that aside from the Anglicans and, more tenuously, the Lutherans, most protestant church congregations are headed by a dude they either elected themselves – the same way Popes are – as Presbys do, or someone they employed from the phone book, the person who assembled that congregation in the first place, or the person who evangelised the locals and forged by suffering the congregation … and aside from the phone book thing, all are methods the earliest churches used to appoint their bishops, teachers and deacons. In each case, their protty prayers would have been the same prayers for God to direct their choice as the Conclave uses, in purpose if not form at least, so, really, if the same Holy Spirit informs both, how are they so different?
God did not come in human form to establish a renewed hierarchical temple with a reinvigorated priesthood to renew the old corrupted one, he came to tear it down and replace it with Himself as head priest, and with believers' bodies as the Holy of Holies, that is, the seat of the Spirit of God. To wit, we believers are the royal priesthood, we are all now the ones who are supposed to be doing the priestly duties, not hiding in lay organisations waiting for a priest to fart us into action. Hierarchy and rigid memberships to a specific organisation and successions are not what Christ was establishing, and certainly not what the Book of Acts or any of Paul's letters describe.
So, you can see from this, OP, that the "problem of legitimacy" is a paper tiger in this protestant's eyes. The legitimacy of any congregation (I don't say "church" for there is not 100,000 "churches" but one Church) is rooted in the Spirit that exists in the congregants' hearts as true believers whom Christ "knows". Now, you may say, "But the host, user, the hoooost. If not one of His own priests celebrates the Eucharist, how will Christ know where to turn wafer into flesh?" … and there we can have an entirely different discussion. Suffice to say that if the Elders of a congregation anoint and lay-on of hands a leader from amongst them, and they have the same Spirit, they are repeating the same pattern the New Testament describes. We are not a religion of very carefully performed rituals, we are a spiritual union with our God. He knows whom He has called out from the world.
But, look, I get it, these are controversial ideas for some people. If such loose ideas about who is "in" and who is "out" of the Church is too discomforting, by all means, join an apostolic church and cling to the idea that your membership, that your name on the church roll is what grants you your election into salvation, not the Spirit who lives within you, and not the fact that Christ can say, "You, in life, I knew. Welcome into your rest."
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