I would be much more interested in UA's LGA-VIE service. Quite a technological feat given the airport's operating limits.

Connections which involve a change of airport do create a need for better communication. The stock answer is that on a connection, one's bags are transferred to the connecting flight unless the connection time exceeds the carrier's holding limit, e.g. 12 hours, overnight or whatever.

NYC is one of very few places where US carriers sell connections between the three airports and that deserves an asterisk in the training materials and perhaps a special warning on at ticketing.

Nonetheless, OP has his answer.