Logic duplication in SearchRequestContext and TimeProvider

[ansjcy]

Describe the bug

SearchRequestContext and TimeProvider both has Phase Took Map, and they are basically recording the same thing. We can investigate if we can improve these logic to make them simpler. Also, we should seek ways to make the workflow to add a SearchOperationsListener easier so that plugins outside of core can also utilize it to create their own listeners.

Related component

Search:Query Insights

To Reproduce

Check the logic in

• https://github.com/opensearch-project/OpenSearch/blob/2.x/server/src/main/java/org/opensearch/action/search/TransportSearchAction.java#L358
• https://github.com/opensearch-project/OpenSearch/blob/2.x/server/src/main/java/org/opensearch/action/search/AbstractSearchAsyncAction.java#L436

They are updated with the same data.

Expected behavior

Remove this duplication by refactoring the code to improve the maintainability and simplicity of the core search path.

Additional Details

Plugins
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Host/Environment (please complete the following information):

• OS: [e.g. iOS]
• Version [e.g. 22]

Additional context
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[ansjcy]

Closing this issue with resolution in #11526