I don't think it had anything to do with ranks and such. Talk is cheap so there's alot of it. I think every Captain, or officer placed in the captain's seat, took stock of the situation and tried to know about every other ship that was in on the encounter, and which fellow captains they would be fighting with. Im sure every other ship's sensors detected the Enterprise arriving and some of the captains were probably great with it, while others might've been worried.
Then the admiral's ship is destroyed and without skipping a beat Picard is on the airwaves saying he has command and to target all of their weapons onto specific coordinates. All of the other captains, whether they were worried or not, had to follow suit as the blown up admirals plan of taking as many pot shots as you could at the thing wasn't working.
Sure you could go back and ask why this happened or why that happened. But as the situation played out it called for the unusual to occur. These people are Starfleet Captains, working "on the fly" is about 80 percent of their job. The decision came down to; Picards here with that new ship we've heard about telling us to fire at these coordinates vs. Picard's here saying he's in command, should we follow?