There is some truth in what you say, but I will have to disagree with your conclusion. It is true that the gospel writers did not sign their work. This was the custom of virtually all ancient writing. If we cancel out all information which is not directly attributed to a named author, then we will have to ignore nearly all ancient histories as “hearsay,” for the simple reason that all or nearly all ancient histories were published anonymously, and only attributed to authors by others.
About the four gospels, like I said, they were published anonymously, which was the custom at the time. But, do we know who wrote them? How would we know? How would we know the author of the accounts of Roman historian Tacitus? Because those who came after them tell us that Tacitus wrote his histories. In the case of the four gospels, it is the virtually unanimous testimony of all who wrote on the subject, including authors in the early second century that Mark wrote Mark, Matthew wrote Matthew, Luke wrote Luke and John wrote John. Luke is an actual person in his two-book sequence Luke/Acts. As far as I know, no serious scholars doubt that Luke wrote Luke. Luke was a very careful historian who clearly interviewed a number of eye-witnesses. Would his testimony be admitted in a court of law? Perhaps not, but as serious and reliable history it is first-rate. I recently read a history of the life of Abraham Lincoln. It was not written by an eye-witness, of course. Does that mean that the history book is “hearsay” and unreliable? I do not think so. If we do not accept the historical accounts of Luke because he was not an eye-witness, then we also reject virtually all history books ever written, as Luke had access to and interviewed eye-witnesses to the events. Matthew and John, obviously, were eye-witnesses to the gospel events. Mark was an eye-witness to at least some of the events, as he is a minor character in his book, and it is well known that he was a close associate of both Paul (earlier) and Peter (later). What better source of accurate information could we have than from Mark, who was part of the inner circle?
As for Matthew, to be fair, it is somewhat less certain that the person we know of as Matthew wrote this gospel. I would say that it is likely the eye-witness Matthew wrote this book, but the evidence is a bit slimmer, and it is not unreasonable to propose that Matthew did not write this book. In any case, unless you can show evidence that the testimony of Matthew or perhaps someone else we do not know is unreliable, then it is not unreasonable to conclude that Matthew’s account is from an eye-witness, probably, although not certainly, from Matthew. As for John, some have questioned if it is the apostle John or an elder closely associated with John who wrote this gospel. Papias mentions the two possibilities in the early second century, but the consensus of the early church was that it was written by the apostle John. The way he calls himself “the beloved disciple” implies to me that it was in fact John who wrote this gospel. I would not bet my life that John wrote John, but the evidence says that it is very likely that John wrote John.
Do you have any evidence that it was NOT John or Matthew who wrote these gospels? You write as if you have some sort of data which proves these are not eye-witness accounts, but do you have any actual evidence that they were not written by the authors they were attributed to in the early second century? Do you have any countervening evidence which shows what Mark wrote is unreliable? You speak above as if everyone knows that John did not write John, but have you looked at the evidence he did write this book? Do you have actual solid evidence that he did not write the book? To be honest, I doubt that you do.
You talk about “quintuple hearsay.” I will have to say that this is the height of hyperbole. You mention Papias, but Papias said that Luke wrote Luke. Where are the five levels of hearsay here? Seriously! Papias allows for the possibility of one level of “hearsay” as you would call it, in that an associate of John, rather than John himself wrote the account. OK. Fine. I believe that John wrote John, rather than another Presbyter John, but even so, this would be written by a close associate of the actual eye-witness, which is nothing even close to quintuple hearsay. I got a laugh out of your hyperbole in this case. My conclusion is that the four gospels were written either by actual eye-witnesses, or by very close associates of the actual eye-witnesses to the gospel events. The level of reliability in this case is at least as good as histories we accept as reliable in our modern context. Of course, you will decide what you think about this, but this is how I see the evidence.
I hope this helps.