Did Jesus exist?

[Minimalist]

Actually, that passage was entirely unknown until it was claimed by Sulpicius Severus in the early 5th century.

Nope, not quite but you have put your finger deftly on the problem.

Here's what Severus wrote ( Book II, 29  Chronica) and he did not cite Tacitus as a source.

In the meantime, the number of the Christians being now very large, it happened that Rome was destroyed by fire, while Nero was stationed at Antium. But the opinion of all cast the odium of causing the fire upon the emperor, and he was believed in this way to have sought for the glory of building a new city. And in fact, Nero could not by any means he tried escape from the charge that the fire had been caused by his orders. He therefore turned the accusation against the Christians, and the most cruel tortures were accordingly inflicted upon the innocent. Nay, even new kinds of death were invented, so that, being covered in the skins of wild beasts, they perished by being devoured by dogs, while many were crucified or slain by fire, and not a few were set apart for this purpose, that, when the day came to a close, they should be consumed to serve for light during the night. In this way, cruelty first began to be manifested against the Christians. Afterwards, too, their religion was prohibited by laws which were enacted; and by edicts openly set forth it was proclaimed unlawful to be a Christian. At that time Paul and Peter were condemned to death, the former being beheaded with a sword, while Peter suffered crucifixion. And while these things went on at Rome, the Jews, not able to endure the injuries they suffered under the rule of Festus Florus, began to rebel.

And here is Annales 15:44 as we have it today.

Consequently, to get rid of the report, Nero fastened the guilt and inflicted the most exquisite tortures on a class hated for their abominations, called Christians by the populace. Christus, from whom the name had its origin, suffered the extreme penalty during the reign of Tiberius at the hands of one of our procurators, Pontius Pilatus, and a most mischievous superstition, thus checked for the moment, again broke out not only in Judæa, the first source of the evil, but even in Rome, where all things hideous and shameful from every part of the world find theircentre and become popular. Accordingly, an arrest was first made of all who pleaded guilty; then, upon their information, an immense multitude was convicted, not so much of the crime of firing the city, as of hatred against mankind. Mockery of every sort was added to their deaths. Covered with the skins of beasts, they were torn by dogs and perished, or were nailed to crosses, or were doomed to the flames and burnt, to serve as a nightly illumination, when daylight had expired.

While the opening of the passage is slightly re-written from the original Severus the red-highlighted part - which is all the jesus freaks care about - is not so much as hinted at by Severus and it would have served him well with the point he was trying to make.  If it had been there.

What is clear is that Severus was writing at a time, early 5th century, when xtians were starting to pull their "oh, poor, poor, persecuted us" bullshit.  That's what he cares about.

But you are quite right when you say that no xtian or pagan writer prior to Severus had ever heard or associated Nero with persecuting xtians for the Great Fire.  Given the oft-stated position that Tacitus was such a "great" historian it seems highly unlikely that the early church writers could have been so negligent as to miss this little tidbit.