As I understand, the GE90 was a complete redesign but did is it fundamentally a pretty similar power plant?
No, GE90 is a clean sheet design compared to the CF6 - about all they have in common design wise is they are both two spool and made by GE.
There have been a number of issues with burst turbine discs on the CF6, but I don't recall the -80C2 variant being affected. The big problem has been with the CF6-50, with a few other well known events on the CF6-80A. Two -80A disc failures occurred during ground runs - one ground run failure resulted in the 767 burning to the ground
(USAir IIRC).
But as noted, I don't recall any such issues with the CF6-80C2 (which was a major redesign of the -80A, not a whole lot in common)