Even if Edmure sucks by whatever standard, and even if he doesn't have the rights to sit on the Iron Throne, you don't just humiliate the veteran Lord Paramount of the Riverlands in front of other Lord Paramounts of Westeros like that. It's a meeting to discuss the next ruler. Let the man talk. If you don't agree with whatever he was going to say, just disagree after he's done. That's basic manner. Hell, we don't even know what Edmure was actually going to say. What if he wanted to support someone else, and that "I'm experienced" talk was just an intro to give weight to his opinions? That was what I thought he was going for in that moment. We don't know, because Sansa cut him off.
And again, this was supposed to be a discussion. Telling other people to shut up means you don't care about what others have to say. In that case, why even have a discussion?
Who the hell is Sansa compared to Edmure anyway? She "won" Winterfell by asking the Vale army, who are foreigners, to occupy it for her. She didn't even command that army. She never fought a battle. She spent the entire story being a political captive in King's Landing, was a suspect for regicide, hid in the Vale like a coward, got herself involved again in the murder of Lysa Arryn, the whole brouhaha with Ramsay Bolton, ran away again, and then came back. Then suddenly they're pushing her as this competent leader who has done... what exactly? She only managed Winterfell for a while, and off-screen. It's not that darn amazing to manage a castle. Freaking crippled child Bran did it. She showed no tact for politics as she kept undermining Jon's authority by questioning him in front of his bannermen. She tried to shoo away a very important ally Jon barely managed to get by telling them to piss off because they're short on food, again, in front of everyone.
And what was Sam even doing there?