And you know this how?
ONLY a bishop can give holy orders for priesthood.
And on what day did that first Bishop do that? The Apostolic Succession is a wonderful thing. To claim one knows exactly how it came about is a bit far-fetched. I would imagine that the earliest assistants of the Apostles as named in the NT were the first bishops ... but then again I imagine.
The record keeping was just fine prior to Constantine - that is why we have the bible today along with many other non-biblical documents and papal letters.
Miraculously indeed we have the New Testament, but the official version of today took a while to develop. But to claim that Church records of the first 3 centuries "are fine" is simply preposterous. Surely, there are some records, perhaps even enough to give a fairly accurate chronology of the papacy, but let's not push our luck!
BTW, scholars of church history have plenty of differences of opinion about these matters, none of which are matters of dogma.
By definition you can’t just convert to be a Catholic priest. You need to receive Holy Orders by a bishop. Below is an example from Acts of a diaconate ordination, but you get the point:
And the plan met the approval of the whole multitude, and they chose Stephen, a man full of faith and of the Holy Spirit, and Philip and Prochorus and Nicanor and Timon and Parmenas and Nicholas, a proselyte from Antioch. These they set before the Apostles, and after they had prayed they laid their hands upon them.
(Acts 6:5-6)
Obviously you have not read Jimmy Akin’s book on Church Father quotes. It is not a small book. There are many letters and documents from the first 300 years of the Church. It is Roman history that is often suspect at various times as they were not good at hiding or copying their records (they didn’t think they needed to) so they often got burned during various sackings. The Church, on the other hand, copied their documents and spread them to the various churches and that is why the record is so well preserved today.