### 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
final ThreadPool threadPool,
        final ResourceWatcherService resourceWatcherService,
        final ScriptService scriptService,
        final NamedXContentRegistry xContentRegistry,
        final Environment environment,
        final NodeEnvironment nodeEnvironment,
        final NamedWriteableRegistry registry,
        final IndexNameExpressionResolver resolver,
        final Supplier<RepositoriesService> repositoriesServiceSupplier,
        TelemetryProvider telemetryProvider,
        AllocationService allocationService,
        IndicesService indicesService
    ) {
        final List<Object> components = new ArrayList<>();
        this.repositoriesServiceSupplier = repositoriesServiceSupplier;
        this.threadPool.set(threadPool);
        this.failShardsListener.set(new FailShardsOnInvalidLicenseClusterListener(getLicenseState(), clusterService.getRerouteService()));
        if (DiscoveryNode.canContainData(settings)) {
            final CacheService cacheService = new CacheService(settings, clusterService, threadPool, new PersistentCache(nodeEnvironment));
            this.cacheService.set(cacheService);
            final SharedBlobCacheService<CacheKey> sharedBlobCacheService = new SharedBlobCacheService<>(
                nodeEnvironment,
                settings,
                threadPool,
                SearchableSnapshots.CACHE_FETCH_ASYNC_THREAD_POOL_NAME
            );
            this.frozenCacheService.set(sharedBlobCacheService);
            components.add(cacheService);
            final BlobStoreCacheService blobStoreCacheService = new BlobStoreCacheService(
                clusterService,