I am assuming that 16: 5 is making the point that as a senator, Tacitus had access to Acta Senatus, the records of the senate. I don't think this means that for every incident or person described, Tacitus would say that he got the information from the Acta.
On the other hand, maybe it was all forged by medieval monks, the cunning little dogs.
Well that would be just multiple speculation, and with zero evidence to support it.
Eg, it's speculating that (1)Jesus was indeed executed by Pilate, (2) that written Roman records did exist describing that execution of Jesus, (3) that those written records were available to Tacitus, (4) that Tacitus did actually use those records for what he later wrote about
"Chrsitsus". But where, afaik, there is zero evidence of any such written Roman records describing the execution of Jesus, and equally zero evidence of Tacitus ever using, or even ever claiming to have seen or used any such records for his one very brief mention of the death of
"Christus".
So that is just a whole heap of un-evidenced uncorroborated speculation.
What is not speculation, however, is that according to bible scholars, Paul's letters and the gospels were already in circulation by the time Tacitus wrote about
"Christus" (and certainly before the earliest existing copies of Tacitus, which are from as late as the 11th century!). And iirc the name of Pilate occurs first of all in one of the
"fake" Pauline letters ... and then it later appears in the earliest relatively complete and readable copies of the gospels which date from about the 4th-6th century onwards (what any first century gospels originally said, is anyone’s guess).
In which case, Paul’s letters (whoever wrote them) and those gospels, together with what Christians themselves were probably saying on the streets at that time, are apparently a known source from which Tacitus and/or his later copyists could have very easily and rather obviously obtained the belief that Jesus was executed under Pilate c.30AD ... a fact which of course, Tacitus himself could not personally have known about, since he was not even born at that time.