The other answers overlook something very neat and simple: the Starks were the last to speak.
Sansa deliberately waited until all the others had said "Aye". Everyone (including most viewers, I imagine) assumed she would say "Aye" to her own brother becoming king - but she waited until everyone else had committed before making her move.
This was very smart. Bran's position was essentially confirmed, so her little power play wouldn't stop Bran being king, and it meant that if anyone else had wanted to copy her, they would have to publicly contradict themselves and publicly go against their own word.
She waited until they had all committed to accepting Bran as their king before putting the idea of breaking away into their heads and revealing it was something Bran would likely agree to in order to maintain peace. There were clear shots of her watching the others, smiling confidently, looking around making sure everyone else had spoken, waiting for the right moment.
She learned a lot about politics from Littlefinger and Cersei over the years...
Of course, if there had been someone like Tywin Lannister present, he would not have been caught flat-footed like the others were. He'd have smelled someone consolidating power and he'd have known that he had about five seconds between Sansa making this play and Bran accepting it to put Sansa's ambitions down.
He'd probably have quickly made the calculation that Sansa cares more about peace and having a mild king like Bran than she cares about independence. He'd have used his extensive political wits to quickly think up a powerful way of putting Sansa under the pressure of thinking that the whole deal would fall through if she didn't back down. He'd also have found a way of presenting it without making enemies of the others present, risking them allying against him in future, and without appearing to have gone back on his word, risking appearing weak and losing credibility.
Edmure Tully, Gendry, Royce, Yara, Sam, Davos... these are not characters with the same political instincts and wits, or ambitions.
Of those present, only Tyrion could realistically have responded effectively here, but he's not going to bluff bringing down his own suggestion, when it looks like his last chance to save himself from Grey Worm's dagger.
It would be perfectly in character for, say, Yara to sail home kicking herself that if she'd played her cards better, maybe she might have wrangled independence for the Iron Islands. It would not be in character for her to magically transform into a master politician and masterfully improvise her way to it in a matter of seconds, or take big risks in a clumsy, improvised power grab when, unlike Sansa, she had been perfectly happy supporting Queen Dany.