Is my plan for mounting a child safety gate feasible?
The metal brackets near the points of your red arrows are intended to be mounted to the wall and act as hinges for the entire gate. I would turn them 90° to the gate and mount them to the green cabinet where you had planned on using the screws. This will allow the gate to swing in one direction at least. You may not be able to swing it into the kitchen as it will bind. At least the gate will function as it was designed.