It is rather bizarre, I agree. The pilot could easily have communicated directly with Delta maintenance on the ground at that point from either JFK or LGA, and I have heard patches to maintenance on those terminal frequencies. The only reason I can offer is that ARINC 129.9 works anywhere offshore from the Canadian Maritimes down to Chesapeake Bay, communicating through multiple zones via eight simulcast, slightly offset frequency transceivers (the reason those ground stations sound so crappy) AND the ARINC operators have all the phone numbers for their airline customers at their fingertips and can put the calls through immediately. The real question is why those ground stations sound like crap, and my own personal theory is that they 1) use speech processing at the ground operator console that is highly compressed and optimized for HF sideband communications in order to punch through noise and static not generally present on VHF and 2) perhaps cross-link their transmitter repeaters via daisy-chain, the audio getting progressively worse with each hop.
If you listen carefully to the ground transmissions on the 129.9 feed, the majority which are from JFK and we believe a few coming from Southampton fifty miles to the east, you will note that some of the transmissions, the ones I believe to be via repeater or cross-link, have about a one second dead carrier tail after the operator un-keys, often heterodyning with the subsequent pilot transmission if he keys up immediately. That heterodyne represents the -2.5kc offset of the JFK transmitter, which is actually transmitting on 129.8975. Southampton transmits 2.5kc higher on 129.9025. Cape Cod and Chesapeake are +7.5 and -7.5 repectively. Apparently airborne receivers certified for the ARINC system use synchronous detection and filtering to lock onto the strongest carrier received and notch out any heterodyne, which allows essentially single channel operation over a very large area.
The same system is used for the terrestrial ARINC coverage, for example 129.4 covers from the NE triangle from Wisconsin down to Chesapeake and up to Maine using a dozen ground stations and the same type of frequency offset simulcast scheme. In a triplet of recordings I posted almost two months ago here: http://www.liveatc.net/forums/atcaviation-audio-clips/typical-sfo-arinc-traffic/
I was able to reconstruct both sides of the conversations thanks to ground station archives recorded out of Boston, which is at present the only feed that captures any 129.4 transmissions from the ground. The aircraft were received at JFK from hundreds of miles away but could not be heard in Boston, and it is doubtful given their positions that ARINC was receiving them via Boston either, since the two of the three flights were almost over-top ARINC stations in Buffalo and Williamsport, and the third was halfway between Williamsport and another ARINC station in Baltimore. Thanks to the simulcast all the ground transmissions were repeated and captured in Boston.