The most accurate prophecies are made after the events. What the prophecy does is bestow the event with an aura of fate, destiny, divine edict, legitimate authority.
The Gospels inform us that Jesus was the prophesied messiah. This itself is not evidence, however, that early first century Jews were generally expecting a messiah as a fulfilment of some ancient scripture.
A Roman emperor, Augustus Caesar, was lauded as a prophesied saviour. Virgil placed his own age, with the advent of Augustus, as the fulfilment of a divinely inspired prophecy, in his fourth eclogue. He again shows Augustus was prophesied from ancient times in book 8 of the Aeneid. I doubt that Romans had been generally longing for Augustus with such prophesies on their lips during the period of the civil war that preceded and led to his rise to power. But after Augustus was in power, Virgil’s poems and epic praises of him found a very receptive public audience.
I know of no evidence that Jews of the early first century were any different from Romans in their expectations and focus during punishing times — the Jews being subject to Roman rule and the Romans to civil war. When one side or group found peace (or ‘peace’ through a form of spiritual escape from reality), that peace — the new order, the new institution — was legitimized, and given comforting assurance, through timely prophecies. Christianity went overboard with this technique and hijacked a whole collection of books from the Jews, declaring their exclusive function was to prophesy of their Saviour.