[Search] Use new elasticsearch-js client in data plugin search service

[lukasolson]

Related: #35508

Now that #69905 is merged, we have access to the new elasticsearch-js client (which updates Typescript support for ES as well as a handful of other benefits), we should update the data plugin search service to use the new client rather than the legacy client. This would open up the ability for other enhancements (see #70675).

[elasticmachine]

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

[pgayvallet]

Now that #69905 is merged, we have access to the new elasticsearch-js client

Note that #69905 only exposes the client internally to core. You'll have to wait until all subtasks of #35508 are addressed before being able to actually use it from plugins. This should be ready at the early beginning of 7.10 hopefully.

[lukasolson]

I believe this has been taken care of at this point.