Actually, the biggest single obstacle to accepting Jesus as the Messiah is 'why him?'
To be fair to the Temple authorities, dozens of people had claimed to be the Messiah, gathered followers - and then mostly tried to lead a revolt.
To the minds of the Jewish authorities, ANY claim to Messiahhood was an incitement to revolt, and had to be suppressed quickly lest the Romans crushed what vestiges of Jewish authority was left.
Of course, they were also aware that their status and authority would be the first to go.
Now, it's true that a less corrupt and political leadership might have at least entertained the possibility that Jeshua bar Joseph was who he claimed to be. The very fact that he was NOT rising in revolt at least separated him from most of the other self-proclaimed messiahs. But, suppose you are on the Sanhedrin. As a body, you think there's a 10% chance he's for real. However, there's a 99% chance that the Romans will oust (and likely execute) you and devastate the whole country if you support him. What would YOU do, honestly?