### 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
client,
                SNAPSHOT_BLOB_CACHE_INDEX
            );
            this.blobStoreCacheService.set(blobStoreCacheService);
            clusterService.addListener(
                new BlobStoreCacheMaintenanceService(settings, clusterService, threadPool, client, SNAPSHOT_BLOB_CACHE_INDEX)
            );
            components.add(blobStoreCacheService);
        } else {
            PersistentCache.cleanUp(settings, nodeEnvironment);
        }
        if (DiscoveryNode.isMasterNode(environment.settings())) {
            
            
            
            
            var usageTracker = new SearchableSnapshotsUsageTracker(getLicenseState(), clusterService::state);
            threadPool.scheduleWithFixedDelay(usageTracker, TimeValue.timeValueMinutes(15), threadPool.generic());
        }
        this.allocator.set(new SearchableSnapshotAllocator(client, clusterService.getRerouteService(), frozenCacheInfoService));
        components.add(new FrozenCacheServiceSupplier(frozenCacheService.get()));
        components.add(new CacheServiceSupplier(cacheService.get()));
        if (DiscoveryNode.isMasterNode(settings)) {
            new SearchableSnapshotIndexMetadataUpgrader(clusterService, threadPool).initialize();
            clusterService.addListener(new RepositoryUuidWatcher(clusterService.getRerouteService()));
        }
        return Collections.unmodifiableList(components);
    }
    @Override
    public void onIndexModule(IndexModule indexModule) {