Fire24

"County Coordinators are called out any time a cause and origin (aka: arson) investigation request is made, and acts as a liaison between the County and the local department, and provides any additional assistance if requested by the on-scene IC"
"The "Battalions" are Mutual Aid Coordinators. They are dispatched anytime Mutual Aid is sent (fire incidents), not including Dual Responses. They're also dispatched whenever a DES resource, such as C&O, Haz-Mat, TRT, etc. is requested to an incident."
The official title for the position is Deputy Fire Mutual Aid Coordinator and there are twelve positions of this type. The term "Battalion" came about years ago under former Commissioner Patrick Kelly so as to have an area of responsibility for the Deputy Fire Coordinators assigned to the mutual aid function. Battalions 10-19 serve in this capacity and, as John noted so well, they are dispatched any time mutual aid or other DES resource is sent to a fire department. The other two "Battalions" are Deputy Fire Coordinators assigned to the Cause & Origin Team, however their identifiers are not Battalions, but Cars. Cars 24 & 25 are essentially supervising fire investigators, one each for the northern and southern portions of the county. C&O is requested by the fire chief to assist them in determining the cause and origin of any fire in their jurisdiction. It's not just for arson, although this is the primary aspect in the protocols for the C&O Team.
The info on the County's website could be more clear on this matter.

There are differences in many things as you move from region to region in the United States; income, accents, history, culture, politics, demographics, etc., etc., etc. The question you raise is not merely applicable to law enforcement. There are vast differences between New York and Texas. I don't think the differences are as great as you believe nor do I think you properly articulated your argument without disparaging cops in NY and the northeast.
I think there is something to be said for the anti-police, anti-government sentiment in the northeast being a contributing factor to your supposition. I think as the country tried to become more politically correct and public service changed from being a calling to a secure job with benefits, we lost some of the service focus that contributed to law enforcement being what it used to be. There are still many hard-chargers in civil service but there are now an equal number of deadbeats just studying for promotional exams with no concept of the job and/or using the stability to advance their education for their next job.
You're flat out wrong about some things. Cops in ESU aren't trying to be firefighters and who says that rescue is a fire function? ESU predates many of the FD rescue resources in Westchester County. Yonkers E-rigs of the 80's were the first paramedics in the city and had the only extrication equipment for many years. The County PD ran an ESU that provided extrication services all over the county before a lot of FD's got their own hurst tools. ESU is not the same as SWAT either. The LAPD runs SWAT but they don't do most of the things that ESU cops do - it is an apples and oranges comparison. In Jersey City, extrications are performed by EMS not fire or PD. White Plains has a police / fire special operations command that works collaboratively.
Duplicating resources? Are you kidding me? You really think that ESU in the couple of cities that have them in Westchester are the duplicative resources and squandering the funding that exists for emergency services. Hmmm, the 59 fire departments, 43 police departments, 60 or so school districts are definitely not duplicative. Who says that EMS is a fire function? PD is an first responder and in many places they are the only BLSFR that responds. We've already covered in other threads that BLSFR is not strictly limited to PD or FD exclusively.
OK, who decides what the core responsibilities are? What do you do when an agency can't/won't/doesn't fulfill it's core responsibilites? Competition exists on many levels in public service and it can be healthy. There are plenty of examples of intra-disciplinary rivalries that are unhealthy; you're broad sweeping generalizations about inter-disciplinary relationships and responsibilities are way off base.
Pity there isn't a negative rep button anymore. Comparing patrol cops in Texas with ESU cops in NY or asserting that certain functions don't have cross-over between disciplines is way off base. There should be collaboration and less division and that's not strictly a PD/FD thing.