I think instinct played an important role here.
You know all that stuff about the primitive brain and all that?
I think the ATC's brain noticed the conflict at an insctintive level, you know, probably with a shiver in his spine, and then the "secondary" brain reacted promptly.
The tone he used was, inmho, perfect. Because probably activated the LH pilot's instinctive brain, too. I bet that they would have rejected take off exactly the same if the ATCo had said "coffe with milk, too sugar" using the same tone.
It is like the naughty kid who always ignores his mothers "don't do this, dont do that, get out of there, drop that..." but who instinctively obeys when the mother voice is genuenily alarmed.
that is communication, too.
On the side of the pilots, it looks very cool to be cool even in a very bad scenario, like an engine burning and severe hydraulic problems and stay calm with a professional, manly tone. But in other scenarios an alarmed voice can do a better job. Sully's "unable" and other communications had an excellent tone, he looked to be in control of the situation, managing it with an adequate stress level (a high one). JFKs ATCo averted a possible reedition of Tenerife, which would have occurred in a matter of seconds. His tone was just what it was required. they did great job, in that. they have to improve their communications, though, and be more international-minded. the macho attitude of "if you can't manage JFK comms, you don't have the right stuff" is a wrong attitude.
hopefully he saw, understood and averted the danger
by the way, i assume there was a take off clearance. Why are the recordings missing that? I will not trust any internet recording anymore