[data.search.esaggs] Move 'other' bucket & inspector integrations to SearchSouce.

[lukeelmers]

Currently the esaggs expression function uses the remnants of what used to be called the "courier request handler", which handles calling SearchSource with the correct aggs.

At the moment it also does some magic like updating the inspector adapters with the request/response data, and also calls an agg postFlightRequest if one exists (for 'other' buckets on terms aggregations).

This means that folks using esaggs are getting the benefit of inspector integration & other bucket handling, but consumers of SearchSource are not. As esaggs is only meant to be an expression function that wraps SearchSource and returns tabified data, we should move this logic to SearchSource entirely:

• Update search source to handle calling agg.postFlightRequest
• Update search source to provide request/response data to the inspector adapters
• Remove this logic from the esaggs request handler

For more context, see the comments on #83199

[elasticmachine]

Pinging @elastic/kibana-app-arch (Team:AppArch)