[Backport] [2.x] Added missed API visibility annotations for public APIs and enable the check at the build time (#12872)

[reta]

Backport of #12872 to 2.x

[github-actions]

Compatibility status:

Checks if related components are compatible with change ef13253

Incompatible components

Skipped components

Compatible components

Compatible components: [https:/opensearch-project/custom-codecs.git, https:/opensearch-project/flow-framework.git, https:/opensearch-project/observability.git, https:/opensearch-project/neural-search.git, https:/opensearch-project/cross-cluster-replication.git, https:/opensearch-project/security-analytics.git, https:/opensearch-project/job-scheduler.git, https:/opensearch-project/geospatial.git, https:/opensearch-project/k-nn.git, https:/opensearch-project/notifications.git, https:/opensearch-project/asynchronous-search.git, https:/opensearch-project/reporting.git, https:/opensearch-project/ml-commons.git, https:/opensearch-project/sql.git, https:/opensearch-project/common-utils.git, https:/opensearch-project/index-management.git, https:/opensearch-project/anomaly-detection.git, https:/opensearch-project/alerting.git, https:/opensearch-project/performance-analyzer-rca.git, https:/opensearch-project/security.git, https:/opensearch-project/performance-analyzer.git]

[github-actions]

❌ Gradle check result for 6b43c73: FAILURE

Please examine the workflow log, locate, and copy-paste the failure(s) below, then iterate to green. Is the failure a flaky test unrelated to your change?

[reta]

Depends on #12867

[github-actions]

❌ Gradle check result for ef13253: ABORTED

Please examine the workflow log, locate, and copy-paste the failure(s) below, then iterate to green. Is the failure a flaky test unrelated to your change?

[github-actions]

❕ Gradle check result for ef13253: UNSTABLE

• TEST FAILURES:

      1 org.opensearch.search.SearchWeightedRoutingIT.testStrictWeightedRoutingWithCustomString_FailOpenEnabled
      1 org.opensearch.repositories.azure.AzureBlobContainerRetriesTests.testWriteLargeBlob
      1 org.opensearch.backwards.MixedClusterClientYamlTestSuiteIT.test {p0=indices.get_field_mapping/20_missing_field/Return empty object if field doesn't exist, but index does}

Please review all flaky tests that succeeded after retry and create an issue if one does not already exist to track the flaky failure.

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 71.09%. Comparing base (0dd892c) to head (ef13253).
Report is 79 commits behind head on 2.x.

Additional details and impacted files

@@             Coverage Diff              @@
##                2.x   #12875      +/-   ##
============================================
- Coverage     71.28%   71.09%   -0.19%     
- Complexity    60145    60386     +241     
============================================
  Files          4957     4985      +28     
  Lines        282799   284400    +1601     
  Branches      41409    41582     +173     
============================================
+ Hits         201591   202207     +616     
- Misses        64189    65118     +929     
- Partials      17019    17075      +56     

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[reta]

@peternied may I ask you please to approve this backport? thank you