Move consumers off bsearch endpoint

[lukasolson]

Summary

Part of https://github.com/elastic/kibana/issues/186139.

Moves direct consumers off of the internal/bsearch endpoint and onto the internal/search.

This is in preparation for removing the internal/bsearch endpoint entirely.

Updates automated tests to mock/use the correct service moving forward.

Checklist

• [ ] Documentation was added for features that require explanation or tutorials
• [ ] Unit or functional tests were updated or added to match the most common scenarios
• [ ] Flaky Test Runner was used on any tests changed