You sound like a good troubleshooter, so I hear you saying...
ESC #3 is where your problem always lies, move motor and ESC pair to another location and they both work (any kind of spin is working), and whatever you put on #3 ESC position it no worky?
the FC tells the ESC when to go, stop, and how much through the signal wire connection, other than that all it needs is good voltage and current from whatever Power Distribution Board (PDB). I would reflow all the possible joints dealing with FC and PDB side, and if that dont fix it the it is the FC or we're gonna learn something new.
So what is your PDB look like (cold solder joint possibly?)
would you have access to a O'scope to see if there is any thing to be seen diff between good esc signal and bad?
Just a few thoughts
Randy