I can see how having redundant services is convenient and helpful when you need them, it just seems like they need to define their duties better; being the first on scene should not be the determining factor over who has control. We have the same problem in my department, separate volly EMS and fire and we both have heavy rescues with Jaws. Basically whoever gets there first does the cutting and runs the scene. While this system can (and generally does) work if the departments get along, you run the risk of situations where each responder is trying to beat the next department to run the Jaws and get IC set up. Everyone needs a defined role that they assume once all of the proper resources are on scene.
It's easy to pick apart the actions of the NYPD/FDNY after the fact but from what I can see it's a matter of command and control- if both agencies are on scene, someone needs to have the authority to say "I'm in charge, this is who it's going to go down." At the same time, the guys already on scene need to have the humility and discipline to swallow their pride and take a less "glamorous" job, e.g. cribbing, PT stabilization, etc. It looks like there was little coordination between agencies and ESU just wanted to get the PT out. If he was DOA, critical, or fine, it doesn't matter- you need to do the job right.