The arguments ended because everyone noticed Banner was holding the Scepter and were unnerved by that. Hulk may have absorbed the power, I highly, highly doubt it though, but that isn’t why everyone stopped arguing.
Notice throughout the moments prior to the reveal everyone has already stopped arguing and Fury and Black Widow have moved their hands to their guns? That’s because they’ve all noticed he’s picked the Scepter up and they are worried, especially knowing what he can become: the Other Guy.
Bruce Banner: In case you needed to kill me, but you can't! I know! I tried! I got low. I didn't see an end, so I put a bullet in my mouth and the other guy spit it out! So I moved on. I focused on helping other people. I was good, until you dragged me back into this freak show and put everyone here at risk!
Banner slowly gets upset as he looks at Romanoff, who gets unnerved.
Bruce Banner: You wanna know my secret, Agent Romanoff? You wanna know how I stay calm?
BLACK WIDOW and Fury have their hands down to grab their guns.
Steve Rogers: Doctor Banner... put down the scepter.
BANNER LOOKS DOWN AND IS SHOCKED TO SEE HE'S HOLDING LOKI'S SCEPTER. The computer beeps. They all turn to it.
Note that Hulk isn’t the only angry one, they are both angry. That is Banner’s secret as he later tells Steve.
Steve Rogers: Dr. Banner. Now might be a really good time for you to get angry.
Bruce Banner: That's my secret, Captain. I'm always angry.
The top answer on the linked question seems to explain why Bruce picks the Scepter up very well. He’s angry and can’t control his emotions and lot happens in a short period of time which gets him worked up leading to him picking up the Scepter.
The presence of the Mind Stone inside the Scepter manipulates those around it. Bruce and Tony has spent time studying it and so were most affected by it leading to Bruce picking the Scepter up on top of the emotional aspect above.