What's wrong with this power latch design?
It's bad design. You press the button and this initially activates the mosfet. That mosfet then turns the bjt on and before you have thought about removing your finger the voltage that was turning the mosfet on (via the pushbutton) has crashed to ground due to the bjt.

Then, a micro second later the mosfet is off and so is the bjt and the process repeats and becomes an oscillator while your finger is pressed.

When you eventually remove your finger the mosfet has a fifty fifty chance of being on. However this won't help because the gate will discharge to ground via the 100 kohm and the circuit turns off.

Note that you cannot expect the sources voltage to be able to drive the gate to hold that mosfet on. Gate has to be several volts above source to fully or partially turn on the mosfet.