Query timeout fails for suggest and rescore phases #122186
Description
The _search request timeout enforces an early termination for queries exceeding their allocated time budget. This timeout is applied per shard and only affects the query phase, ensuring that a partial response can still be returned when triggered.
However, PR #98715 modified this behavior by removing the timeout restriction for the aggregation build phase but did not account for the suggest and rescore phases. As a result, queries that exceed their timeout now throw an unexpected exception (time_exceeded_exception) when running additional phases that are not setup to handle timeouts.