### 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
Setting.Property.IndexScope,
        Setting.Property.PrivateIndex,
        Setting.Property.NotCopyableOnResize
    );
    public static final Setting<String> SNAPSHOT_INDEX_NAME_SETTING = Setting.simpleString(
        "index.store.snapshot.index_name",
        Setting.Property.IndexScope,
        Setting.Property.PrivateIndex,
        Setting.Property.NotCopyableOnResize
    );
    public static final Setting<String> SNAPSHOT_INDEX_ID_SETTING = Setting.simpleString(
        "index.store.snapshot.index_uuid",
        Setting.Property.IndexScope,
        Setting.Property.PrivateIndex,
        Setting.Property.NotCopyableOnResize
    );
    public static final Setting<Boolean> SNAPSHOT_CACHE_ENABLED_SETTING = Setting.boolSetting(
        "index.store.snapshot.cache.enabled",
        true,
        Setting.Property.IndexScope,
        Setting.Property.NotCopyableOnResize
    );
    public static final Setting<Boolean> SNAPSHOT_CACHE_PREWARM_ENABLED_SETTING = Setting.boolSetting(
        "index.store.snapshot.cache.prewarm.enabled",
        true,
        Setting.Property.IndexScope,
        Setting.Property.NotCopyableOnResize
    );
    
    public static final Setting<List<String>> SNAPSHOT_CACHE_EXCLUDED_FILE_TYPES_SETTING = Setting.stringListSetting(