### 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<IndexMetadata> indexMetadataList,
        List<IndexRoutingTable> indexRoutingTables,
        List<NodeShutdown> nodeShutdowns,
        List<DiscoveryNode> nodes
    ) {
        var routingTableBuilder = RoutingTable.builder();
        for (IndexRoutingTable indexRoutingTable : indexRoutingTables) {
            routingTableBuilder.add(indexRoutingTable);
        }
        var nodesShutdownMetadata = new NodesShutdownMetadata(
            nodeShutdowns.stream()
                .collect(
                    toMap(
                        it -> it.nodeId,
                        it -> SingleNodeShutdownMetadata.builder()
                            .setNodeId(it.nodeId)
                            .setType(it.type)
                            .setReason("test")
                            .setNodeSeen(true)
                            .setStartedAtMillis(System.currentTimeMillis())
                            .setAllocationDelay(it.allocationDelaySeconds != null ? timeValueSeconds(it.allocationDelaySeconds) : null)
                            .build()
                    )
                )
        );
        Metadata.Builder metadataBuilder = Metadata.builder();
        Map<String, IndexMetadata> indexMetadataMap = new HashMap<>();
        for (IndexMetadata indexMetadata : indexMetadataList) {