Had this dead GameCube sitting on the shelf for a while, and decided to take one last go at it before writing it off as a lost cause.
The problem was that near one of the RAM modules, one of the vias had corrosion on the inside. This made the via on the back side of the board not have connection with the pad on the other side of the board.
I've already tried to fix this in the past and could not solder to the pad on the front side of the board - it was under the memory chip, if only slightly. Today I looked up some pictures of the an exposed GC memory die to see if I could scrape off a bit of the corner of the chip to get to that particular pad.
I wasn't entirely sure, but it does look empty there (top right corner), so I chipped some of it off, just to find the pad I was trying to blindly solder to to be completely rusted away.
So, in a last ditch effort I tried to scrape away at the board to expose the trace.
I managed to solder to SOMETHING
View attachment 434711
And, to my complete astonishment, the GC fired up.
View attachment 434712
...the reliability of this connection is dubious at best, though, unfortunately :v