The answer is a Hell YES. Here the second supervisor stays outside the hot zone to be the tac coordinator. He or she runs the show. Not the first supervisor who goes into the hot zone, regardless of rank( politics aside).
So a hell yes, to that question. And if you don’t think NYPD isn’t going to take over the incident from some local department inside NYPD jurisdiction like the Sea Gate police or Co-op city police, you are crazy. Yes, that Lowly NYPD sergeant is going to takeover that scene from that sea gate police chief. If it happens inside a state courthouse, the jurisdiction of court officers, hell yes, that lowly sergeant is taking over.
Taking over a s-show with a school to keep kids from dying. Hell, yes.
That SWAT team was outside waiting outside even before the search for the key should say something. The IC or tac commander outside the hot zone would have known they were available a lot earlier.
And if they believed that door was locked and they lacked proper breeching tools or training , the shooting had stopped and someone ( the school chief) declared it a hostage situation, you might wait for SWAT. A SWAT team was there and waited for the word to go in, then waited for a key. That’s when that border patrol team said, we have to go in and did. If the tac coordinator was outside where he or she knew they had SWAT available, they would have been sent in earlier. The school chief didn’t have radio communication, remember.