Why gate drive transformer is not connected directly to the gates in this App Note?
Without digging into the application note itself, I would think that the gate driver U2 and its local power supply are used to get higher current (faster edges) for switching the MOSFETs than you could get from the transformer alone.

This would help with the overall efficiency of the circuit. Note that when either transistor is switched on, it's carrying the full output current (i.e., the current through both inductors).