CorrectionProducers computing should handle errors differently

[FMorschel]

@bwilkerson's #60683 (comment)

[W]e should guard the execution of CorrectionProducers such that any exception thrown causes the correction to not be produced, but doesn't prevent other correction producers from running and results from being returned.

Opening this issue to track that work. I do have a question on how to handle this if we want to still get stacktraces for wrong cases?

[bwilkerson]

The AnalysisServer has an InstrumentationService that supports a logExceptionResult method. The exception should be logged using that method when it's caught.