90 years isn’t a long time and many contemporary primary and secondary sources no doubt existed. If I was speaking about US history, I could mention Gov Al Smith of New York contemporaneously and many people with context would understand me immediately. Tacitus wasn’t writing for an audience a millennia in the future.
He was an imperial official and solider who sympathized with the barbarians resisting the empire. So it wouldn’t be surprising if he used sources outside of the mainstream of the day. I think the ultimate answer is that we don’t know who his source was. If you look at the entirety of independent sources mentioning Christ, I think most scholars agree that he was a historical figure… but the details of what he did when and why do not have that level of consensus.
Tacitus not being a Christian makes him not automatically an "independent source" in the eyes of historians. We need to ask, what was most likely his source (and the source of his source, etc.). This is why historians often prefer to use in such circumstances the more precise term "independent tranditions" instead of "independent sources". Two "independent sources" are either an independent primary or secondary (or tertiary, or ...) source unconnected to each other. Discussing what speeks for or against the independence of sources about the same topic is bread-and-butter work of so-called "source criticism".
That we do not know the source of Tacitus' knowledge and probably never will is not unusual, but the rule for the ancient world. We only have a few important primary sources from that era.[1]
What a historian of Antiquity learns during her or his training and what makes a large part of their academic work is to evaluate such secondary (or tertiary or more remote) sources regarding their most likely history of tradition. In this respect Tacitus' account is a very bad candidate of being really independent, since it is extremely unlikely that it was not even indirectly influenced by someone who knows something about "Christ" without ever having come into contact with the Christian tradition. If instead, he would have claimed that his knowledge was based on old documents from the archives of Pontius Pilate, the picture would be entirely different. Though, we still could have our doubts that such archives were really authentic and not Christian "forgeries".[2]
[1] We have almost no autographs of important historic persons from Antiquity. A noteworthy exception are letters from Bar-Kokhba, the leader of the Jewish revolt from 131-136. See: https://en.wikipedia.org/wiki/Cave_of_Letters#Bar-Kokhba_let...
[2] Christians produced a lot of fictional document, such as letters of Jesus and the Apostels, some of which made it into the New Testament. See: https://en.wikipedia.org/wiki/Pseudepigrapha#Biblical_studie...
Personally, for what it’s worth, I think Christ was a real person whose life and death created a cult of personality at a key moment in time. The details of the story have been fiddled with 1000 times over a millennia and are not reliable historically. Entire branches of Christianity have vanished with varying beliefs as to the details of the story.
Tacitus writes aprox. 90 years after the event. What were his sources? Christianity seems to have already achieved a certain notoriety, from which Tacitus quickly distances himself ("a most mischievous superstition"). But the story fits well to support his anti-imperial and specifically anti-Neronian stance. Therefore we do not need to expect a lot of deep research on his side before including this little remark into his Annals. And the mentioning of Pontius Pilatus sounds suspiciously canonically Christian. The most likely conclusion is that Tacitus is just repeating what his contemporary Christians have told everyone about the origin of their cult.
This does not mean that what he tells us is incorrect, but that is account is probably based on a Christian and not an independent tradition.