### 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
false,
            UnassignedInfo.AllocationStatus.NO_ATTEMPT,
            Collections.emptySet(),
            null
        );
    }
    private static UnassignedInfo decidersNo() {
        return new UnassignedInfo(
            UnassignedInfo.Reason.ALLOCATION_FAILED,
            null,
            null,
            1,
            0,
            0,
            false,
            UnassignedInfo.AllocationStatus.DECIDERS_NO,
            Collections.emptySet(),
            null
        );
    }
    private record ShardRoutingKey(String index, int shard, boolean primary) {}
    private static ShardsAvailabilityHealthIndicatorService createShardsAvailabilityIndicatorService() {
        return createShardsAvailabilityIndicatorService(ClusterState.EMPTY_STATE, Collections.emptyMap());
    }
    private static ShardsAvailabilityHealthIndicatorService createShardsAvailabilityIndicatorService(ClusterState clusterState) {
        return createShardsAvailabilityIndicatorService(clusterState, Collections.emptyMap());
    }
    private static ShardsAvailabilityHealthIndicatorService createShardsAvailabilityIndicatorService(
        ClusterState clusterState,
        final Map<ShardRoutingKey, ShardAllocationDecision> decisions