Interesting. We agree that we have no rotation about the longitudinal (roll) axis in the turn, right*?
This leaves us with the pitch axis and the yaw axis on which to project the components of the turning motion, the axis of which is vertical.
For no yaw and with the roll axis removed from the equation, the entire rotation of the turn has to be around the pitch axis.
But the rotation is around a vertical axis. Ergo, the pitch axis has to be vertical which implies a 90-degree bank.
Does your definition of yaw differ from mine? I e “rotation about the yaw axis of the aircraft”.
Regards,
Fred
*) Not quite true unless the longitudinal axis is level, but let’s assume a flight condition providing this.
(Typo edit)