### 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
new ScalingExecutorBuilder(
                CACHE_PREWARMING_THREAD_POOL_NAME,
                0,
                16,
                TimeValue.timeValueSeconds(30L),
                rejectAfterShutdown,
                CACHE_PREWARMING_THREAD_POOL_SETTING
            ) };
    }
    private static Settings getIndexSettings() {
        return Settings.builder()
            .put(IndexMetadata.SETTING_NUMBER_OF_SHARDS, 1)
            .put(IndexMetadata.SETTING_AUTO_EXPAND_REPLICAS, "0-1")
            .put(IndexMetadata.SETTING_PRIORITY, "900")
            .put(IndexSettings.INDEX_TRANSLOG_DURABILITY_SETTING.getKey(), Translog.Durability.ASYNC)
            .put(DataTier.TIER_PREFERENCE, DATA_TIERS_CACHE_INDEX_PREFERENCE)
            .build();
    }
    private static XContentBuilder getIndexMappings() {
        try {
            final XContentBuilder builder = jsonBuilder();
            {
                builder.startObject();
                {
                    builder.startObject(SINGLE_MAPPING_NAME);
                    builder.field("dynamic", "strict");
                    {
                        builder.startObject("_meta");
                        builder.field("version", Version.CURRENT);
                        builder.field(SystemIndexDescriptor.VERSION_META_KEY, SEARCHABLE_SNAPSHOTS_INDEX_MAPPINGS_VERSION);