The fuse was connected to the battery (the note BAT1). Why is the fuse unnecessary?Whatever he's doing, he needs the pull up voltage to be high enough to turn the MOSFET on. 1.25V isn't high enough to guarantee the device will turn on. The threshold voltage can be as high as 1.45V. 4V will give a typical on resistance of 20-30 milliohms.
Also the pullup is connected right to the battery so the voltage on the gate of the fet should be 4.2V - 3.8V