Is a Cessna 172S still airworthy if factory-installed instrumentation not required for day VFR flight is inoperative?
Part 91 does not require an EGT probe for any operations so the next thing to look at is your AFM.

Your C172S has an AFM that includes a Comprehensive Equipment List. It is probably similar to this one that indicates that the EGT probe is standard equipment—not required for certification, this airplane can fly with it placarded.