This "hot-off-the-press" hypothesis can't be substantiated because we don't have any of these supposed copies nor do we have any report of them. That the Church finally settled on just FOUR gospels, however, suggests that these got into circulation comparatively early and were widely circulated. Commerce and correspondance moved around the Med. in the First Century about as fast as it did it in the 19th Century. I don't rule out that they were available, at least in some form, during the first generation, since modern speculations--which put most of them into the 70-85 period seem to me to be just that--speculations.
It is probably more likely that limited copies were made by the chuch receiving the letter for local communities (this may account for variant endings for certain epistles...Originals would have greetings to individuals within the local church...copies sent to other towns might not) and that more copies spread as churchs shared what they had been given. I understand that this was the case for much of Jewish Scripture as well (the dead sea scrolls do not contain precisely the same "canon" that we use today).
That is untrue, there were I believe over 6000 manuscripts of the NT found, complete, partial, and fragments, and most of these were those copies that each Church has made when an epistle was finisned.