### 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.index.shard.ShardId;
import org.elasticsearch.index.translog.Translog;
import org.elasticsearch.index.translog.TranslogStats;
import org.elasticsearch.indices.IndicesService;
import org.elasticsearch.indices.SystemIndexDescriptor;
import org.elasticsearch.license.LicenseUtils;
import org.elasticsearch.license.XPackLicenseState;
import org.elasticsearch.plugins.ActionPlugin;
import org.elasticsearch.plugins.ClusterPlugin;
import org.elasticsearch.plugins.EnginePlugin;
import org.elasticsearch.plugins.IndexStorePlugin;
import org.elasticsearch.plugins.Plugin;
import org.elasticsearch.plugins.SystemIndexPlugin;
import org.elasticsearch.repositories.RepositoriesService;
import org.elasticsearch.repositories.Repository;
import org.elasticsearch.repositories.RepositoryData;
import org.elasticsearch.repositories.blobstore.BlobStoreRepository;
import org.elasticsearch.rest.RestController;
import org.elasticsearch.rest.RestHandler;
import org.elasticsearch.script.ScriptService;
import org.elasticsearch.snapshots.SearchableSnapshotsSettings;
import org.elasticsearch.snapshots.sourceonly.SourceOnlySnapshotRepository;
import org.elasticsearch.telemetry.TelemetryProvider;
import org.elasticsearch.threadpool.ExecutorBuilder;
import org.elasticsearch.threadpool.ScalingExecutorBuilder;
import org.elasticsearch.threadpool.ThreadPool;
import org.elasticsearch.watcher.ResourceWatcherService;
import org.elasticsearch.xcontent.NamedXContentRegistry;
import org.elasticsearch.xcontent.XContentBuilder;
import org.elasticsearch.xpack.core.XPackPlugin;