### 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
@Nullable
        private final CacheService cacheService;
        CacheServiceSupplier(@Nullable CacheService cacheService) {
            this.cacheService = cacheService;
        }
        @Override
        public CacheService get() {
            return cacheService;
        }
    }
    /**
     * Allows to inject the {@link SharedBlobCacheService} instance to transport actions
     */
    public static final class FrozenCacheServiceSupplier implements Supplier<SharedBlobCacheService<CacheKey>> {
        @Nullable
        private final SharedBlobCacheService<CacheKey> sharedBlobCacheService;
        FrozenCacheServiceSupplier(@Nullable SharedBlobCacheService<CacheKey> sharedBlobCacheService) {
            this.sharedBlobCacheService = sharedBlobCacheService;
        }
        @Override
        public SharedBlobCacheService<CacheKey> get() {
            return sharedBlobCacheService;
        }
    }
    private static final class RepositoryUuidWatcher implements ClusterStateListener {