Is 'hot-wiring' two contacts a proper way to bypass a broken button?
You can bypass the switch in the way you suggest.

You would need to temporarily connect the switch's Pin 1 to a GND node.  Any ground connection should work; you don't need to use any of the specific connections soldered to the switch.  Also, no need to connect the grounds together.

On the switch, Pin 1 and Pin 2 get connected when the button is pressed (obviously).  Pins 3, 4, and 5 are connection to the switch's chassis, and are provided to ground out the switch case.  This acts as a shield for EMI.

Be warned: soldering to a cell phone motherboard can be very tricky.  It's easy to make it worse instead of better.  Also, as mentioned in the comments, the short wires may act like antennas at cell frequencies.  I expect this won't be a problem, but it could cause strange stuff to happen.

If you decide to proceed, I recommend placing the phone in Airplane Mode as soon as you can.