[8.10] Update error handling for saved query service (#163904)

[kibanamachine]

Backport

This will backport the following commits from main to 8.10:

• Update error handling for saved query service (#163904)

Questions ?

Please refer to the Backport tool documentation

[kibana-ci]

💛 Build succeeded, but was flaky

• Buildkite Build
• Commit: 2cb3911

Failed CI Steps

• Serverless Security Defend Workflows Cypress Tests

Metrics [docs]

Public APIs missing comments

Total count of every public API that lacks a comment. Target amount is 0. Run node scripts/build_api_docs --plugin [yourplugin] --stats comments for more detailed information.

id	before	after	diff
data	2575	2576	+1

Page load bundle

Size of the bundles that are downloaded on every page load. Target size is below 100kb

id	before	after	diff
data	404.4KB	404.5KB	+55.0B

Unknown metric groups

API count

id	before	after	diff
data	3301	3302	+1

To update your PR or re-run it, just comment with:
@elasticmachine merge upstream

cc @lukasolson