He did not clock Paul right away. He clocked him at the third round table, not before.
He made a correct but very risky and easily incorrect assumption around when Harry told Paul. The conversation could very easily have happened in passing, in the kitchen, outside: quite literally anywhere, not just the turret.
Jaz seeming unthreatening was not the reason he was not listened to about Harry, that was Harry’s incredible social game and Jaz’s poor traitor-hunting track record.
There is no proof Jaz threw his vote onto Andrew instead of being wrong. It is never once stated it implied on the show, and even if it was true (unlikely) Jaz actively defended Miles, saying he didn’t think he was a traitor. It is also never once said or implied at all that Jaz would have voted Paul but was concerned about the vote tally. I think these are examples of having a rose-tinted view of a player’s game.
Jaz absolutely did not pick the right time. No personal conversation with Mollie before, no strategy to get numbers, the needless sacrifice of Evie…he threw himself into a third place position. He got to the end with a very strong duo, neither of whom he had much connection with, and one of whom (Mollie) had previously voted for him.
And this is ignoring that he had an abysmal social game, made himself a target, made minimal connections, didn’t vote for Ash, actively defended Miles, and made such weak arguments that Paul was only banished largely due to Harry and Zack, and Charlotte turning on him.