[Meta] Batched Query Phase Follow-up Tasks

[original-brownbear]

Quick follow up tasks mostly around tests for after #121885 lands, covering what isn't tracked in the rollout doc:

• Fix tests running rest-api-spec/src/yamlRestTest/resources/rest-api-spec/test/search/120_batch_reduce_size.yml
• Adjust SearchErrorTraceIT and remove batched execution override (Fix SearchErrorTraceIT and friends to work with batched query execution #127150)
• Adjust TasksIT and remove batched execution override
• Adjust SearchCancellationIT and remove batched execution override
• Adjust TermsDocCountErrorIT to be useful with batched execution
• Adjust tests in SearchCancellationIT and remove batched execution override
• Adjust tests in AsyncSearchErrorTraceIT and remove batched execution override (Fix SearchErrorTraceIT and friends to work with batched query execution #127150)
• Optimise failure path if no partial results are requested in org.elasticsearch.action.search.SearchQueryThenFetchAsyncAction#executeShardTasks

[elasticsearchmachine]

Pinging @elastic/es-search-foundations (Team:Search Foundations)