I can't ignore an imminent failure in a component in a kiln when I can't be certain that the failure won't lead to overheating.
For industrial controls, we address risks like that in a few ways:
1) Avoid single points of failure. In some cases, this means redundant sensors.
2) Scheduled preventive maintenance - components with finite lifetimes (i.e. thermocouple probes) are replaced regularly.
3) Design the system to 'fail safe'. In the case of temperature controls, in the absence of credible feedback, you bring the system to a safe state. Electric kilns are fairly straightforward systems in that regard. Things get much trickier when you are dealing with combustible gasses.
are you saying that noise can cause these errors to report?
Yes, noise can cause these types of errors. The sensor is simply looking at voltages on pins. Errors are generated when certain thresholds are exceeded. The 'human antenna' test of pinching the probe tip demonstrates that nicely,
Why would OP's problem resolve by grounding the breakout board to the kiln?
Noise couples into systems in a variety of ways - some of which are not at all obvious. For example: if noise radiated from the kiln is picked up inductively by the probe, there will be voltage fluctuations at the inputs to the amp. If you ground the amp to the source of the voltage fluctuations, the signal are measured relative to that - cancelling out the noise.