[Search] Remove legacy ES client

[lukasolson]

In #53766 we are exposing an instance of the Elasticsearch JavaScript client in the __LEGACY namespace for the purposes of continuing to support the usage of _msearch until bfetch is integrated into our search services (to avoid running into the browser connection limit).

This, and a couple of other places in Kibana are the only remaining usages of esClient, including the following:

• index-pattern creation wizard
• timelion?
• vega?

These places should be migrated to use our data plugin search service. At that point we can remove this esClient service entirely, as well as abortableRequestHandler, since this is the last place it is used.

Related: #54147, #54821

[elasticmachine]

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

[mshustov]

We need to remove elasticsearch-browser workaround from kbn-ui-shared-deps as well

[joshdover]

@lukasolson Looking at the code, it appears that any plugin using the data.search.searchSource.* APIs are still dependent on this legacy client, not just timelion and vega. Is that correct? Are there any blockers to moving to bfetch?

[lukasolson]

That's correct. I would just add that the searchSource APIs only use this legacy client when the courier:batchSearches advanced setting is turned on (and it's turned off by default). No blockers, just a matter of priority at this point.

[mshustov]

It would be great to get rid of this several Mb eagerly loading dependency sooner than later.