There are a few components involved in charging the Mac and working off of battery and/or charger.
SMC is part of that circuit but my assumption would be more towards the "charging IC" and some resistors around it and some "chips" that charge the battery, together with their traces on or through the board.
If you want to explore this, you will need to specify if the green light is on when a charger is plugged in. And what does the battery show: "not charging" or "x" etc.
Then you need to provide us with the board number, find the schematics and board view based on that number, and you will need a multimeter with fine probes to test the lines and the components. And some form of 10x-20x magnification.
But you will reach a point where you will need to remove/replace parts. And that's not a DYI job.
Following the update you posted, my opinion is that the ISL chip, a few resistors, a couple of traces are potentially damaged. It should be totally fixable at ½ the cost of a new board. Shop around and do the math.
If you need to find someone who can fix it let me know where you are so that I can recommend someone.