[Index Patterns] Switch to the new Elasticsearch client

[Dosant]

Part of #83910

Old client is still used in functional tests:

test/api_integration/apis/index_patterns/es_errors/errors.js:23:
test/functional/apps/management/_create_index_pattern_wizard.js:14:
test/functional/apps/management/_handle_alias.js:13:
test/functional/apps/management/_handle_version_conflict.js:24:
test/functional/apps/management/_index_patterns_empty.ts:18:
x-pack/test/functional/apps/management/create_index_pattern_wizard.js:10

[elasticmachine]

Pinging @elastic/kibana-app-services (Team:AppServices)