[cj1]

If you have good bat then test for intermittent start circuit.

Since its so intermittent you may have to use a test light that you can see while behind the wheel when starting engine.

Get a electrical diagram of the starting circuit.
First step determine if starter solenoid is getting voltage during no crank.
Connect wire from solenoid to test light.

If no voltage during no crank than follow diagram and connect light to next logical test point.
If 12v then suspect starter if connections/wiring from bat good.

 

[cj1]

Since it is so intermittent best to have test light near driver. This will mean running wire from solenoid to inside cab.

The click you hear may be start relay or starter solenoid. Verify that solenoid is/or is not getting power during the no crank condition.

Of course if you can catch the problem by testing numerous times with hood open, test light on solenoid, so much the better.

 

[cj1]

"Can anyone explain why disconnecting the battery for second let's it start up as if nothing happened?"

Do you have a remote start or after market anti-theft that may be resetting with the disconnect of the battery?

"In videos on youtube of impalas with bad starters it's almost like a passlock issue, except in my case all the electronics go dead!"

Regarding passlock, security light is normal?
Electronics go dead. Does this mean dash lights go out when trying to start?

Take a look at Eric the Car Guy no crank no start video.

 

[cj1]

If you can verify that the solenoid is getting correct voltage during the no crank then suspect intermittent starter problem.
This assumes that bat can give you at least 10v during cranking and that starter circuit wiring/connections are good.

 

[cj1]

Just for grins,try disconnecting the opposite bat connection to see if the fox still works. Maybe there is a mechanical/electrical problem with one connector or bat.

 

[cj1]

Any other codes besides the ABS?
Consider having dealer check for BCM codes.

There are a lot of possible problems, as you listed, but without a schematic and volt meter to narrow down possibilities its change parts and hope for the best.

Keep in mind that in the process of changing parts, installation and faulty parts can cause addition problems.