@SamBushman
It looks you have put in a lot of effort in checking the traces!
One thing to bear in mind is that sometimes the issue is not that the trace is broken from A to B where it is supposed to go, but because it could also be shorted to C. Thus, your continuity test between A and B will always show continuity between them, but it does not show continuity to C, unless you measure from A to C.
What you could do is to try to measure for any shorts to either ground or VCC. The could be for those traces that have pull down resistors to ground or pull up resistors to VCC.
Since you have DiagROM but your serial output is garbage then I would try to sort that out first - so check around the serial chips. Without output to screen and/or serial, DiagROM is useless. It is common that these serial chips, or components and traces associated with them fail and they need to be repaired/replaced.
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