It think that it's generally overlooked that Jellico was brought in to do exactly what he did. He had his mission, just like Picard had one, & neither circumstance was ideal, but for the situation, they were the ideal candidates. That's orders. You take them & do them. Who you are don't matter.
Jellico even grants Picard a bit of a berth, during his final plea to ease up on his family of officers, & Jellico even agrees that it would be better to have an chance to get to know them & create a trusted bond with them, but that's idealistic crap. That's not what he was given. He got another man's ship & crew, & told to do something with them that they don't normally do. Any captain, put in the same position would have to do the exact same things Jellico did, even Picard. Putting the ENT crew on edge was the only option he had, with the time he had. "Get it done"? that's probably not his slogan on the Cairo. It had to be for this mission though.
I'm certain that the Cairo crew doesn't find him unreasonable at all. He did what he did because that's what needed to be done. He studied the specs, & the crew & the ship, & the mission. He knew everything & everyone & every variable, before he stepped off the transporter pad. He knew right from the start that conflict & friction was inevitable, so he had no chioce but to make that work for him. that was the mission they gave him
Orders trump will, like the scene in Lower Decks, when the junior officers have to shut their friends up, because they can't talk about it. You don't have to like your orders, but you have to follow them, even when they stand in opposition with what you believe. You don't have to like your mission, but you need to do your damnedest to pull it off
This is why I think this episode is the perfect foil to the earlier Cardy episode "The Wounded" where Picard has to keep the peace, at any cost. He acted a way, in that episode, which was not how he would have, acting solely on his own will, & even tells the Cardy that at the end. Jellico was doing the same, but because you did not know him, it is easy to miss. I think it's very smart of TNG to put such a subtle device in their story. It's a story about the subtlety of communication breakdown, a study in the mechanisms we have which form impressions of those around us. A trully brilliant episode. The TNG Cardassian episodes are easily some of the best Trek ever made