### 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
import org.elasticsearch.xpack.searchablesnapshots.rest.RestClearSearchableSnapshotsCacheAction;
import org.elasticsearch.xpack.searchablesnapshots.rest.RestMountSearchableSnapshotAction;
import org.elasticsearch.xpack.searchablesnapshots.rest.RestSearchableSnapshotsNodeCachesStatsAction;
import org.elasticsearch.xpack.searchablesnapshots.rest.RestSearchableSnapshotsStatsAction;
import org.elasticsearch.xpack.searchablesnapshots.store.SearchableSnapshotDirectory;
import org.elasticsearch.xpack.searchablesnapshots.upgrade.SearchableSnapshotIndexMetadataUpgrader;
import java.io.IOException;
import java.io.UncheckedIOException;
import java.util.ArrayList;
import java.util.Collection;
import java.util.Collections;
import java.util.HashSet;
import java.util.List;
import java.util.Locale;
import java.util.Map;
import java.util.Optional;
import java.util.Set;
import java.util.function.Function;
import java.util.function.Predicate;
import java.util.function.Supplier;
import java.util.stream.Collectors;
import static org.elasticsearch.index.mapper.MapperService.SINGLE_MAPPING_NAME;
import static org.elasticsearch.snapshots.SearchableSnapshotsSettings.SEARCHABLE_SNAPSHOT_STORE_TYPE;
import static org.elasticsearch.xcontent.XContentFactory.jsonBuilder;
import static org.elasticsearch.xpack.core.ClientHelper.SEARCHABLE_SNAPSHOTS_ORIGIN;
import static org.elasticsearch.xpack.core.searchablesnapshots.SearchableSnapshotsConstants.SEARCHABLE_SNAPSHOT_FEATURE;
/**
 * Plugin for Searchable Snapshots feature
 */
public class SearchableSnapshots extends Plugin implements IndexStorePlugin, EnginePlugin, ActionPlugin, ClusterPlugin, SystemIndexPlugin {