### 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
indexMetadataMap.put(indexMetadata.getIndex().getName(), indexMetadata);
        }
        metadataBuilder.indices(indexMetadataMap);
        metadataBuilder.putCustom(NodesShutdownMetadata.TYPE, nodesShutdownMetadata);
        DiscoveryNodes.Builder discoveryNodesBuilder = DiscoveryNodes.builder();
        nodes.forEach(discoveryNodesBuilder::add);
        return ClusterState.builder(new ClusterName("test-cluster"))
            .routingTable(routingTableBuilder.build())
            .nodes(discoveryNodesBuilder)
            .metadata(metadataBuilder.build())
            .build();
    }
    private static Map<String, Object> addDefaults(Map<String, Object> override) {
        return Map.of(
            "unassigned_primaries",
            override.getOrDefault("unassigned_primaries", 0),
            "initializing_primaries",
            override.getOrDefault("initializing_primaries", 0),
            "creating_primaries",
            override.getOrDefault("creating_primaries", 0),
            "restarting_primaries",
            override.getOrDefault("restarting_primaries", 0),
            "started_primaries",
            override.getOrDefault("started_primaries", 0),
            "unassigned_replicas",
            override.getOrDefault("unassigned_replicas", 0),
            "creating_replicas",
            override.getOrDefault("creating_replicas", 0),
            "initializing_replicas",
            override.getOrDefault("initializing_replicas", 0),