### 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_
server/src/main/java/org/elasticsearch/snapshots/SearchableSnapshotsSettings.java
false,
        Setting.Property.IndexScope,
        Setting.Property.PrivateIndex,
        Setting.Property.NotCopyableOnResize
    );
    public static final String SEARCHABLE_SNAPSHOTS_REPOSITORY_NAME_SETTING_KEY = "index.store.snapshot.repository_name";
    public static final String SEARCHABLE_SNAPSHOTS_REPOSITORY_UUID_SETTING_KEY = "index.store.snapshot.repository_uuid";
    public static final String SEARCHABLE_SNAPSHOTS_SNAPSHOT_NAME_SETTING_KEY = "index.store.snapshot.snapshot_name";
    public static final String SEARCHABLE_SNAPSHOTS_SNAPSHOT_UUID_SETTING_KEY = "index.store.snapshot.snapshot_uuid";
    public static final String SEARCHABLE_SNAPSHOTS_DELETE_SNAPSHOT_ON_INDEX_DELETION = "index.store.snapshot.delete_searchable_snapshot";
    private SearchableSnapshotsSettings() {}
    public static boolean isSearchableSnapshotStore(Settings indexSettings) {
        return SEARCHABLE_SNAPSHOT_STORE_TYPE.equals(INDEX_STORE_TYPE_SETTING.get(indexSettings));
    }
    public static boolean isPartialSearchableSnapshotIndex(Settings indexSettings) {
        return isSearchableSnapshotStore(indexSettings) && indexSettings.getAsBoolean(SEARCHABLE_SNAPSHOT_PARTIAL_SETTING_KEY, false);
    }
    /**
     * Based on a map from setting to value, do the settings represent a partial searchable snapshot index?
     *
     * Both index.store.type and index.store.snapshot.partial must be supplied.
     */
    public static boolean isPartialSearchableSnapshotIndex(Map<Setting<?>, Object> indexSettings) {
        assert indexSettings.containsKey(INDEX_STORE_TYPE_SETTING) : "must include store type in map";
        assert indexSettings.get(SNAPSHOT_PARTIAL_SETTING) != null : "partial setting must be non-null in map (has default value)";
        return SEARCHABLE_SNAPSHOT_STORE_TYPE.equals(indexSettings.get(INDEX_STORE_TYPE_SETTING))
            && (boolean) indexSettings.get(SNAPSHOT_PARTIAL_SETTING);
    }
}