Follow-up to SERVER-129849.

The idle-mongot-cursor disposal marker is set in run_aggregate.cpp from LiteParsedPipeline::hasMongotStage(), which only inspects top-level stages.

Gap: LiteParsedLookup/LiteParsedUnionWith forward only the extension-stage checks, so a built-in $search/$vectorSearch nested in a sub-pipeline is not detected. A $unionWith whose sub-pipeline contains $search builds that sub-pipeline eagerly and retains the mongot task executor; if the outer cursor is registered with batchSize: 0, OpDebug::mongotCursorId is never set either, so the cursor is not disposed at shutdown and shutdown waits out the searchTaskExecutorShutdownTimeoutMS backstop (default 60s). $lookup builds its sub-pipeline lazily, so it is a narrower case.

Fix direction: make mongot-stage detection recurse through sub-pipelines (single source of truth), consider renaming the marker to mayHoldMongotTaskExecutor to reflect that ownership is inferred rather than ground truth, and add regression tests for a nested $search/$vectorSearch under $unionWith (and $lookup if reachable) with batchSize: 0.