### 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
private final SetOnce<SharedBlobCacheService<CacheKey>> frozenCacheService = new SetOnce<>();
    private final SetOnce<ThreadPool> threadPool = new SetOnce<>();
    private final SetOnce<FailShardsOnInvalidLicenseClusterListener> failShardsListener = new SetOnce<>();
    private final SetOnce<SearchableSnapshotAllocator> allocator = new SetOnce<>();
    private final Settings settings;
    private final FrozenCacheInfoService frozenCacheInfoService = new FrozenCacheInfoService();
    public SearchableSnapshots(final Settings settings) {
        this.settings = settings;
    }
    public static void ensureValidLicense(XPackLicenseState licenseState) {
        if (SEARCHABLE_SNAPSHOT_FEATURE.checkWithoutTracking(licenseState) == false) {
            throw LicenseUtils.newComplianceException("searchable-snapshots");
        }
    }
    public static BlobStoreRepository getSearchableRepository(Repository repository) {
        if (repository instanceof SourceOnlySnapshotRepository) {
            repository = ((SourceOnlySnapshotRepository) repository).getDelegate();
        }
        if (repository instanceof BlobStoreRepository == false) {
            throw new IllegalArgumentException("Repository [" + repository + "] is not searchable");
        }
        return (BlobStoreRepository) repository;
    }
    public static Predicate<CacheKey> forceEvictPredicate(ShardId shardId, Settings indexSettings) {
        final String snapshotUUID = SNAPSHOT_SNAPSHOT_ID_SETTING.get(indexSettings);
        final String snapshotIndexName = SNAPSHOT_INDEX_NAME_SETTING.get(indexSettings);
        return k -> shardId.equals(k.shardId()) && snapshotIndexName.equals(k.snapshotIndexName()) && snapshotUUID.equals(k.snapshotUUID());
    }
    @Override
    public List<Setting<?>> getSettings() {