[Alerting] Remove legacy ES client usage in the alerting functional tests

[YulNaumenko]

Functional tests
x-pack/test/alerting_api_integration/security_and_spaces/tests/actions/builtin_action_types/es_index.ts:16:
x-pack/test/alerting_api_integration/security_and_spaces/tests/actions/builtin_action_types/es_index_preconfigured.ts:18:
x-pack/test/alerting_api_integration/security_and_spaces/tests/actions/execute.ts:26:
x-pack/test/alerting_api_integration/security_and_spaces/tests/alerting/alerts.ts:29:
x-pack/test/alerting_api_integration/security_and_spaces/tests/alerting/create.ts:23:
x-pack/test/alerting_api_integration/security_and_spaces/tests/alerting/delete.ts:22:
x-pack/test/alerting_api_integration/security_and_spaces/tests/alerting/disable.ts:23:
x-pack/test/alerting_api_integration/security_and_spaces/tests/alerting/enable.ts:23:
x-pack/test/alerting_api_integration/security_and_spaces/tests/alerting/rbac_legacy.ts:17:
x-pack/test/alerting_api_integration/spaces_only/tests/actions/builtin_action_types/es_index.ts:16:
x-pack/test/alerting_api_integration/spaces_only/tests/actions/execute.ts:25:
x-pack/test/alerting_api_integration/spaces_only/tests/alerting/alerts_base.ts:26:
x-pack/test/alerting_api_integration/spaces_only/tests/alerting/builtin_alert_types/es_query/alert.ts:35:
x-pack/test/alerting_api_integration/spaces_only/tests/alerting/builtin_alert_types/index_threshold/alert.ts:34:
x-pack/test/alerting_api_integration/spaces_only/tests/alerting/builtin_alert_types/index_threshold/fields_endpoint.ts:20:
x-pack/test/alerting_api_integration/spaces_only/tests/alerting/builtin_alert_types/index_threshold/indices_endpoint.ts:21:
x-pack/test/alerting_api_integration/spaces_only/tests/alerting/builtin_alert_types/index_threshold/time_series_query_endpoint.ts:54:
x-pack/test/alerting_api_integration/spaces_only/tests/alerting/create.ts:22:
x-pack/test/alerting_api_integration/spaces_only/tests/alerting/delete.ts:16:
x-pack/test/alerting_api_integration/spaces_only/tests/alerting/disable.ts:21:
x-pack/test/alerting_api_integration/spaces_only/tests/alerting/enable.ts:21:
x-pack/test/plugin_api_integration/test_suites/event_log/service_api_integration.ts:14:

[elasticmachine]

Pinging @elastic/kibana-alerting-services (Team:Alerting Services)

[mikecote]

This needs to be done by 7.15 feature freeze. If we can't, we should notify the core team (#83910 (comment)).

[mikecote]

We should check event log and task manager as well.