### 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
.build();
        IndexMetadata regularIndex = IndexMetadata.builder("regular-index")
            .settings(Settings.builder().put(IndexMetadata.SETTING_VERSION_CREATED, IndexVersion.current()).build())
            .numberOfShards(1)
            .numberOfReplicas(0)
            .build();
        String featureDataStreamName = ".test-ds-feature";
        IndexMetadata backingIndex = createBackingIndex(featureDataStreamName, 1).build();
        ShardRouting featureIndexRouting = createShardRouting(
            new ShardId(featureIndex.getIndex(), 0),
            true,
            new ShardAllocation(randomNodeId(), UNAVAILABLE, noShardCopy())
        );
        ShardRouting regularIndexRouting = createShardRouting(
            new ShardId(regularIndex.getIndex(), 0),
            true,
            new ShardAllocation(randomNodeId(), UNAVAILABLE, noShardCopy())
        );
        ShardRouting backingIndexRouting = createShardRouting(
            new ShardId(backingIndex.getIndex(), 0),
            true,
            new ShardAllocation(randomNodeId(), UNAVAILABLE, noShardCopy())
        );
        var clusterState = createClusterStateWith(
            List.of(featureIndex, regularIndex, backingIndex),
            List.of(
                IndexRoutingTable.builder(featureIndex.getIndex()).addShard(featureIndexRouting).build(),
                IndexRoutingTable.builder(regularIndex.getIndex()).addShard(regularIndexRouting).build(),
                IndexRoutingTable.builder(backingIndex.getIndex()).addShard(backingIndexRouting).build()
            ),