tests: address warnings in timeline shutdown (#10702)

## Problem

There are a couple of log warnings tripping up
`test_timeline_archival_chaos`

- `[stopping left-over name="timeline_delete"
tenant_shard_id=2d526292b67dac0e6425266d7079c253
timeline_id=Some(44ba36bfdee5023672c93778985facd9)
kind=TimelineDeletionWorker\n')](https://neon-github-public-dev.s3.amazonaws.com/reports/pr-10672/13161357302/index.html#/testresult/716b997bb1d8a021)`
- `ignoring attempt to restart exited flush_loop
503d8f401d8887cfaae873040a6cc193/d5eed0673ba37d8992f7ec411363a7e3\n')`

Related: https://github.com/neondatabase/neon/issues/10389

## Summary of changes

- Downgrade the 'ignoring attempt to restart' to info -- there's nothing
in the design that forbids this happening, i.e. someone calling
maybe_spawn_flush_loop concurrently with shutdown()
- Prevent timeline deletion tasks outliving tenants by carrying a
gateguard. This logically makes sense because the deletion process does
call into Tenant to update manifests.

pageserver/src/tenant/timeline.rs

@@ -2633,7 +2633,7 @@ impl Timeline {
                 return;
             }
             FlushLoopState::Exited => {
-                warn!(
+                info!(
                     "ignoring attempt to restart exited flush_loop {}/{}",
                     self.tenant_shard_id, self.timeline_id
                 );

pageserver/src/tenant/timeline/delete.rs

@@ -341,6 +341,13 @@ impl DeleteTimelineFlow {
         let tenant_shard_id = timeline.tenant_shard_id();
         let timeline_id = timeline.timeline_id();
 
+        // Take a tenant gate guard, because timeline deletion needs access to the tenant to update its manifest.
+        let Ok(tenant_guard) = tenant.gate.enter() else {
+            // It is safe to simply skip here, because we only schedule background work once the timeline is durably marked for deletion.
+            info!("Tenant is shutting down, timeline deletion will be resumed when it next starts");
+            return;
+        };
+
         task_mgr::spawn(
             task_mgr::BACKGROUND_RUNTIME.handle(),
             TaskKind::TimelineDeletionWorker,
@@ -348,6 +355,8 @@ impl DeleteTimelineFlow {
             Some(timeline_id),
             "timeline_delete",
             async move {
+                let _guard = tenant_guard;
+
                 if let Err(err) = Self::background(guard, conf, &tenant, &timeline, remote_client).await {
                     // Only log as an error if it's not a cancellation.
                     if matches!(err, DeleteTimelineError::Cancelled) {