### 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.cluster.metadata.RepositoriesMetadata;
import org.elasticsearch.cluster.metadata.RepositoryMetadata;
import org.elasticsearch.cluster.node.DiscoveryNode;
import org.elasticsearch.cluster.node.DiscoveryNodes;
import org.elasticsearch.cluster.routing.RerouteService;
import org.elasticsearch.cluster.routing.allocation.AllocationService;
import org.elasticsearch.cluster.routing.allocation.DataTier;
import org.elasticsearch.cluster.routing.allocation.ExistingShardsAllocator;
import org.elasticsearch.cluster.routing.allocation.decider.AllocationDecider;
import org.elasticsearch.cluster.service.ClusterService;
import org.elasticsearch.common.Priority;
import org.elasticsearch.common.io.stream.NamedWriteableRegistry;
import org.elasticsearch.common.settings.ClusterSettings;
import org.elasticsearch.common.settings.IndexScopedSettings;
import org.elasticsearch.common.settings.Setting;
import org.elasticsearch.common.settings.Settings;
import org.elasticsearch.common.settings.SettingsFilter;
import org.elasticsearch.common.unit.ByteSizeUnit;
import org.elasticsearch.common.unit.ByteSizeValue;
import org.elasticsearch.common.util.concurrent.EsExecutors;
import org.elasticsearch.core.Nullable;
import org.elasticsearch.core.Releasables;
import org.elasticsearch.core.TimeValue;
import org.elasticsearch.env.Environment;
import org.elasticsearch.env.NodeEnvironment;
import org.elasticsearch.index.IndexModule;
import org.elasticsearch.index.IndexSettings;
import org.elasticsearch.index.engine.EngineFactory;
import org.elasticsearch.index.engine.ReadOnlyEngine;
import org.elasticsearch.index.engine.frozen.FrozenEngine;