Capture liveness and readiness probe errors

The following discussion from !317 (merged) should be addressed:

• @ash2k started a discussion: (+2 comments)

  This is good, but ideally we want to also send it to Sentry. Here and in a few other places we need a log+Sentry wrapper for ease of use like we already have as part of modserver.API. It should be extracted from there for reuse (but not removed from the API, just extracted). Does not have to be part of this MR, just saying that it's needed. Perhaps just leave a TODO.