Even the second flight plan, with the AR
routes, isn't really considered an "overwater" route, since it remains within 162 miles of the shoreline. Most airliners are equipped to fly these routes, since all you really need are life vests. The "overwater" routes take you further away from shore, and therefore require more cabin equipment, like life rafts and survival gear. Many overwater routes also take you out of VHF range and require an HF
radio. This equipment is expensive and many domestic airliners aren't suitably equipped. An "overwater" route from JFK
would use "amber" airways, such as A300 or A557, not just AR
(Atlantic Route) airways, such as AR7. This also means that you'd traverse RVSM airspace, and even though domestic airlines have to be equipped for RVSM by next January, not all operators have a RVSM program currently in place.
Therefore, it's entirely appropriate for ATC to question whether the flight in question is capable of accepting such routes. The benefit of the routes themselves is that they bypass Washington Center, which regularly freaks out when storms approach, and starts shutting down airspace. The amber routes take you away from the mess and on your way. They do take a little longer to fly, but that's far better than waiting for Wash to reopen their airspace.