On paper, it appears Sparrow Essentials Board is compatible with Notecarrier-F due to 3.3V JST SH, which Sparrow uses I2C to search for Notecarrier; correct?
That is correct.
The Sparrow Gateway needs access to a Notecard. This is most easily achieved via a Qwiic connector (but not absolutely mandatory). Therefore, any Notecarrier with a Qwiic connector will work well to construct a Sparrow Gateway.
However, it is not possible to use the Feather socket at the same to interact with the Notecard, because the Notecard is not designed to interact with two hosts simultaneously (e.g. Sparrow and the Feather MCU).
Depending on your needs and/or available hardware, the Notecarrier-F may make a suitable solution for constructing a Sparrow Gateway.