EDIT: Ok, went and re-listened to that episode again. Here's a transcript of the part the OP is referencing.

[Talking about Otho's suicide] "Though his suicide did indeed endear him to future generations, who saw Otho's death as some of the most noble and virtuous in the long history of the Empire, it would only stop the bloodshed for a short while. For many Romans, while Otho may have been a murderous usurper, Vitellius was a crass and unworthy man who used the armies entrusted to him to engage in full blown civil war without soliciting the Senate or the people's approval.

Vitellius may have been popular with his troops, and supported by a few provincial governors in the west, but rest of the Empire was a different story. Backers of Otho were not going to resign themselves to Vitellius; rather, they cast their net looking for a leader to rally behind; someone to dislodge the Rhine legions from their new place at the top of the food chain. Naturally, their eyes turned to the east.

Too far away to really take part in the events of past year, the eastern legions led by Titus Flavius Vespasianus, read about the unfolding chaos with dismay. When Vitellius of all people wound up on the throne, Vespasian decided to heed the call of his fellow eastern governors to bid for the throne himself.

After all, had he not heard the prophecy that the future rulers of the world will emerge from Judea? Embroiled in the Jewish revolt - in Judea - Vespasian had not only heard the prophecy, he had come to believe it. Just a month and a half after his so-called decisive victory, Vitellius discovered that the entire eastern Empire, including the key province of Egypt, had declared for Vespasian. Despite Otho's best efforts, the chaos and bloodshed of the year of the four Emperors was far from over."

EDIT 2:

From Josephus' Jewish War:

"But now, what did the most elevate them in undertaking this war, was an ambiguous oracle that was also found in their sacred writings, how," about that time, one from their country should become governor of the habitable earth." The Jews took this prediction to belong to themselves in particular, and many of the wise men were thereby deceived in their determination. Now this oracle certainly denoted the government of Vespasian, who was appointed emperor in Judea. However, it is not possible for men to avoid fate, although they see it beforehand. But these men interpreted some of these signals according to their own pleasure, and some of them they utterly despised, until their madness was demonstrated, both by the taking of their city and their own destruction."

Suetonius references it as well in Life of Vespasian:

"A firm persuasion had long prevailed through all the East, that it was fated for the empire of the world, at that time, to devolve on some one who should go forth from Judaea. This prediction referred to a Roman emperor, as the event shewed; but the Jews, applying it to themselves, broke out into rebellion, and having defeated and slain their governor, routed the lieutenant of Syria, a man of consular rank, who was advancing to his assistance, and took an eagle, the standard of one of his legions."

EDIT 3:

I found a bit more in "Prophecy in Early Christianity and the Ancient Mediterranean World", via google books: [here] (https://books.google.rs/books?id=0cY3kedl8NMC&pg=PA140&lpg=PA140&ots=TTiNJI6RTo&dq=vespasian+prophecy&hl=en&redir_esc=y)

Tacitus, Hist 3 (sorry this edition doesn't have chapter numbers) says there was a revolt in Britain let by a Venutius, who had a problem with Queen Cartismandua. Venutius, in T.'s opinion, was taking advantage of Roman disorganization to move against Cartismandua. Rome's forces in Britain supported Cartismandua - so they were otherwise occupied. Cartismandua apparently lost.

4.81 (this one does have chapter numbers) discusses the supposed miracles.

[4.81] In the months during which Vespasian was waiting at Alexandria for the periodical return of the summer gales and settled weather at sea, many wonders occurred which seemed to point him out as the object of the favour of heaven and of the partiality of the Gods. One of the common people of Alexandria, well known for his blindness, threw himself at the Emperor's knees, and implored him with groans to heal his infirmity. This he did by the advice of the God Serapis, whom this nation, devoted as it is to many superstitions, worships more than any other divinity. He begged Vespasian that he would deign to moisten his cheeks and eye-balls with his spittle. Another with a diseased hand, at the counsel of the same God, prayed that the limb might feet the print of a Caesar's foot. At first Vespasian ridiculed and repulsed them. They persisted; and he, though on the one hand he feared the scandal of a fruitless attempt, yet, on the other, was induced by the entreaties of the men and by the language of his flatterers to hope for success. At last he ordered that the opinion of physicians should be taken, as to whether such blindness and infirmity were within the reach of human skill. They discussed the matter from different points of view. "In the one case," they said, "the faculty of sight was not wholly destroyed, and might return, if the obstacies were removed; in the other case, the limb, which had fallen into a diseased condition, might be restored, if a healing influence were applied; such, perhaps, might be the pleasure of the Gods, and the Emperor might be chosen to be the minister of the divine will; at any rate, all the glory of a successful remedy would be Caesar's, while the ridicule of failure would fall on the sufferers." And so Vespasian, supposing that all things were possible to his good fortune, and that nothing was any longer past belief, with a joyful countenance, amid the intense expectation of the multitude of bystanders, accomplished what was required. The hand was instantly restored to its use, and the light of day again shone upon the blind. Persons actually present attest both facts, even now when nothing is to be gained by falsehood.

I have no idea what this has to do with Jesus or Palestine; this takes place in Egypt, where Alexander was recognized as the son of a god. I think Suetonius reports him uttering the phrase near death "I think I am becoming a god" which I read as entirely sarcastic. Tacitus is writing this after Domitian's rule, I think, somewhere between 100 and 110 CE, so 30-40 years after the event. The question is who these "persons" are - living when Tacitus is writing? That's what I take the last clause to mean. I rather think his divine claims are geared more toward propaganda to secure his base in Egypt; I don't remember him doing anything like that at Rome. As far as I know Domitian was the first to declare himself a living god in Rome.

I didn't know scholars had a consensus on Vitellius.

Who did this podcast. Would I be wrong to guess Dan Carlin?