I know of no prophet who claims to be teaching a new doctrine that improves upon the past and is disconnected with it
I think all of them do (other than Protestantism). Every one of the other Christian-derived religions has new scriptures that came centuries or more after Christ that offer new and unheard-of revelations.
All a man concerned with the return to the uncorrupted beliefs of the Early Church need do is quote the writings of the Early Church
To Protestants, the "Early Church" is the first-century Church during the lives of the apostles and the writings that they consider reliable from this period are called the "New Testament". Where we disagree is in your assumption that there was a single organization known as the Church in this period. To Protestants, the Church is simply the aggregate body of all believers; it is not an organization and there is no special "approved" set of doctrines that could be checked. Every individual church had its own peculiarities. Again, there simply is no authoritative source other than the New Testament.
and then a long gap where nothing was written and nothing was said worthy noticing, until the rise of the founder of the breakaway Church, whose words are studied with care
If there are earlier writers who don't propound serious theological errors then I'm sure that they would be read with respect if not deference, but the errors of the Catholics are considered very serious by modern Protestants. Taking religious advice from a man who prays to Mary seems as dubious to a Protestant as taking religious advice from a man who prays to Zeus would seem to you. You can't really appreciate the Protestant view without appreciating how appalled they are at Catholic practices. When they accuse Catholics of idolatry and worship of false gods (aka "saints") --that is not just mean-spirited rhetoric; it is a description of their honest position on the matter. And it is not a capricious position but one that is soundly supported by scripture and history.
The older the date on the claim, the less believable it is. I am more willing to believe an argument that the Church of AD 1400 went astray than that the Church of AD 400 or AD 40
Again Protestants don't accept the idea of a single organization called "the Church" so there is no single point of failure to map this to. That is, there is no single time when the Church went astray. The Church has always contended with false doctrines and there have always been false brethren among the true. We know this from the Epistles and from the teachings of Christ who predicted exactly that. You agree with this presumably, but where we differ is that to Protestants the solution to this problem is to test every teacher and prophet in the light of Scripture as Jesus instructed, while you believe that there is a special authority, an organization that does not need to be tested because it is the God-ordained Church. (Actually, I'm not really sure what you think about that because you must be aware that there have been some extraordinarily evil things done by the Catholic Church, so you must agree that the Catholic Church and the Pope cannot be trusted without reservation, right?)
In a game of Russian Telephone, the boy who hears the message first is, statistically speaking, less likely to be suffering from accumulated errors
That doesn't really apply --or only very weakly-- when the communication is written rather than oral. By that consideration, the oral traditions of the Catholic Church are much more questionable than the written scriptures.
It is less unbelievable to say the followers of a student of a disciple of an apostle of Christ mistook or corrupted the teaching of Christ than to say that the disciples of the apostles mistook or corrupted it; still less the apostles; still less Christ Himself.
Only Christ is without error. We know for a fact that the Apostles erred. Peter denied Christ three times. Even in later years, Paul writes of having to correct Peter for hypocrisy. And we know that direct students of the Apostles erred because lots of the Epistles were written to correct them. this idea of some Golden Age of Church Perfection is just not something that Protestants believe, and frankly it's not at all credible.
a mere theologian claims to have deduced the original and uncorrupted teaching of the Church using no other source than official Church teachings, and the reflections of natural reason
And many prayers and supplications for the Holy Spirit to give guidance and wisdom. Protestant theologians and teachers are, for the most part, very humbly aware of the solemn task that they have taken on. They do not have the confidence of a Pope or a Priest that he has been granted a special dispensation to speak for God. They are just regular, error-prone men who have to rely on the grace of God to do anything good because good comes only from God. Really, ask a Protestant preacher some time how he can be so sure he's right and I bet you will be surprised at the humility of the answer. The idea that Protestants rely purely on natural reason is a distortion that, I speculate, comes from conflating the Reformation with the Enlightenment.
If the one, true, catholic, and apostolic Church Christ founded is corrupt and heretical, then Christ is forsworn of his word to send a comforter to guide his disciples in to all wisdom, or, to be precise, that the Church disobeyed this spirit.
I'm not sure how the Church disobeying the Spirit amounts to Christ being foresworn. He often predicted that such things would happen and that's probably why he never created a single hierarchy of leadership for the Church. Instead he created 12 leaders, each to found dozens of churches, each church to send out missionaries to found dozens more. And as the Enemy corrupted one church, other uncorrupted churches would spring up to replace them and carry on the Good Work. To Protestants, the Catholic Church is just one of the Enemy's biggest successes --a mustard seed that grew out of all proportion and the birds of the air (which represented Satan in Jesus's parables) nested in its branches. The Protestants are not without historical justification when they call the Catholic Church the Whore of Babylon --the whore (a people unfaithful to God) who was drunk with the blood of the saints (a Christian, anyone who partakes of the Holy Spirit). The Catholic Church murdered many Christians during her time of dominance --a difficulty that it seems is much more pressing for you than your hermeneutic difficulties are for Protestants.
by what right can that one next claim that the revived church resisted corruption for a season, a decade, or a century or three, or however old it is now?
They don't. There is, in this corrupt age, no single Church, pure and uncorrupted. The wheat will continue to grow with the tares until the final harvest. At most a Protestant would say "I believe that this church, today, is within the will of God because I have tested it and it is true to Scripture." That's why many Protestant churches have what they call a "statement of faith". It is not, as some outsiders believe, intended to teach doctrine, rather it is their test answers, to be handed out to anyone who wants to grade how well they conform to Scripture.