### 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/test/java/org/elasticsearch/cluster/routing/allocation/ShardsAvailabilityHealthIndicatorServiceTests.java
}
    private static ClusterState createClusterStateWith(List<IndexRoutingTable> indexRoutes, List<NodeShutdown> nodeShutdowns) {
        List<IndexMetadata> indices = indexRoutes.stream()
            .map(
                table -> IndexMetadata.builder(table.getIndex().getName())
                    .settings(Settings.builder().put(IndexMetadata.SETTING_VERSION_CREATED, IndexVersion.current()).build())
                    .numberOfShards(1)
                    .numberOfReplicas(table.size() - 1)
                    .build()
            )
            .collect(Collectors.toList());
        return createClusterStateWith(indices, indexRoutes, nodeShutdowns, List.of());
    }
    private static List<IndexMetadata> createIndexMetadataForIndexNameToPriorityMap(Map<String, Integer> indexNameToPriorityMap) {
        List<IndexMetadata> indexMetadataList = new ArrayList<>();
        if (indexNameToPriorityMap != null) {
            for (Map.Entry<String, Integer> indexNameToPriority : indexNameToPriorityMap.entrySet()) {
                String indexName = indexNameToPriority.getKey();
                IndexMetadata.Builder indexMetadataBuilder = new IndexMetadata.Builder(indexName);
                Settings settings = Settings.builder()
                    .put(IndexMetadata.SETTING_PRIORITY, indexNameToPriority.getValue())
                    .put(IndexMetadata.INDEX_NUMBER_OF_SHARDS_SETTING.getKey(), 1)
                    .put(IndexMetadata.INDEX_NUMBER_OF_REPLICAS_SETTING.getKey(), 1)
                    .put(IndexMetadata.SETTING_INDEX_VERSION_CREATED.getKey(), IndexVersion.current())
                    .build();
                indexMetadataBuilder.settings(settings);
                indexMetadataList.add(indexMetadataBuilder.build());
            }
        }
        return indexMetadataList;