### 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/cluster/routing/allocation/ShardsAvailabilityHealthIndicatorService.java
private static final Logger LOGGER = LogManager.getLogger(ShardsAvailabilityHealthIndicatorService.class);
    public static final String NAME = "shards_availability";
    private static final String DATA_TIER_ALLOCATION_DECIDER_NAME = "data_tier";
    private final ClusterService clusterService;
    private final AllocationService allocationService;
    private final SystemIndices systemIndices;
    public ShardsAvailabilityHealthIndicatorService(
        ClusterService clusterService,
        AllocationService allocationService,
        SystemIndices systemIndices
    ) {
        this.clusterService = clusterService;
        this.allocationService = allocationService;
        this.systemIndices = systemIndices;
    }
    @Override
    public String name() {
        return NAME;
    }
    @Override
    public HealthIndicatorResult calculate(boolean verbose, int maxAffectedResourcesCount, HealthInfo healthInfo) {
        var state = clusterService.state();
        var shutdown = state.getMetadata().custom(NodesShutdownMetadata.TYPE, NodesShutdownMetadata.EMPTY);
        var status = new ShardAllocationStatus(state.getMetadata());
        for (IndexRoutingTable indexShardRouting : state.routingTable()) {