### 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
) {
        return createAllocationHealthIndicatorService(Settings.EMPTY, clusterState, decisions, new SystemIndices(List.of()));
    }
    private static ShardsAvailabilityHealthIndicatorService createShardsAvailabilityIndicatorService(
        Settings nodeSettings,
        ClusterState clusterState,
        final Map<ShardRoutingKey, ShardAllocationDecision> decisions
    ) {
        return createAllocationHealthIndicatorService(nodeSettings, clusterState, decisions, new SystemIndices(List.of()));
    }
    private static ShardsAvailabilityHealthIndicatorService createAllocationHealthIndicatorService(
        Settings nodeSettings,
        ClusterState clusterState,
        final Map<ShardRoutingKey, ShardAllocationDecision> decisions,
        SystemIndices systemIndices
    ) {
        var clusterService = mock(ClusterService.class);
        when(clusterService.state()).thenReturn(clusterState);
        var clusterSettings = new ClusterSettings(nodeSettings, ClusterSettings.BUILT_IN_CLUSTER_SETTINGS);
        when(clusterService.getClusterSettings()).thenReturn(clusterSettings);
        var allocationService = mock(AllocationService.class);
        when(allocationService.explainShardAllocation(any(), any())).thenAnswer((Answer<ShardAllocationDecision>) invocation -> {
            ShardRouting shardRouting = invocation.getArgument(0);
            var key = new ShardRoutingKey(shardRouting.getIndexName(), shardRouting.getId(), shardRouting.primary());
            return decisions.getOrDefault(key, ShardAllocationDecision.NOT_TAKEN);
        });
        return new ShardsAvailabilityHealthIndicatorService(clusterService, allocationService, systemIndices);
    }
}