### Elasticsearch Version

8.8.1+ but probably all versions

### Installed Plugins

_No response_

### Java Version

_bundled_

### OS Version

cloud

### Problem Description

This is a follow-up of [#97370](https://github.com/elastic/elasticsearch/issues/97370). 

In a case where disk spaces are reaching limits, if ILM suddenly tries to move a lot of hot shards to cold/frozen shards, it can tip the cluster health to RED because a lot of cold/frozen shards may be created and stay unassigned as long as there is not enough disk space. However, this is not a suitable indicator for service unavailability, since the data is still safely in the respective hot shards that are becoming cold/frozen, and will remain there until the cold/frozen shards are assigned so that searches can still search them.

Thus the [Health Report API](https://www.elastic.co/guide/en/elasticsearch/reference/8.8/health-api.html) should not return RED for unassigned cold/frozen shards of ILM moves.

### Steps to Reproduce

In a case where disk spaces are reaching limits, if ILM suddenly tries to move a lot of hot shards to cold/frozen shards, it can tip the cluster health to RED because a lot of cold/frozen shards may be created and stay unassigned (as long as there is not enough disk space).

### Logs (if relevant)

_No response_
x-pack/plugin/searchable-snapshots/src/main/java/org/elasticsearch/xpack/searchablesnapshots/SearchableSnapshots.java
false,
                        indexSettings.getValue(SourceOnlySnapshotRepository.SOURCE_ONLY)
                            ? SourceOnlySnapshotRepository.readerWrapper(engineConfig)
                            : Function.identity(),
                        false,
                        true
                    )
                );
            }
        }
        return Optional.empty();
    }
    @Override
    public List<ActionHandler<? extends ActionRequest, ? extends ActionResponse>> getActions() {
        return List.of(
            new ActionHandler<>(SearchableSnapshotsStatsAction.INSTANCE, TransportSearchableSnapshotsStatsAction.class),
            new ActionHandler<>(ClearSearchableSnapshotsCacheAction.INSTANCE, TransportClearSearchableSnapshotsCacheAction.class),
            new ActionHandler<>(MountSearchableSnapshotAction.INSTANCE, TransportMountSearchableSnapshotAction.class),
            new ActionHandler<>(XPackUsageFeatureAction.SEARCHABLE_SNAPSHOTS, SearchableSnapshotsUsageTransportAction.class),
            new ActionHandler<>(XPackInfoFeatureAction.SEARCHABLE_SNAPSHOTS, SearchableSnapshotsInfoTransportAction.class),
            new ActionHandler<>(TransportSearchableSnapshotCacheStoresAction.TYPE, TransportSearchableSnapshotCacheStoresAction.class),
            new ActionHandler<>(FrozenCacheInfoAction.INSTANCE, FrozenCacheInfoAction.TransportAction.class),
            new ActionHandler<>(FrozenCacheInfoNodeAction.INSTANCE, FrozenCacheInfoNodeAction.TransportAction.class),
            new ActionHandler<>(
                TransportSearchableSnapshotsNodeCachesStatsAction.TYPE,
                TransportSearchableSnapshotsNodeCachesStatsAction.class
            )
        );
    }
    public List<RestHandler> getRestHandlers(