### 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
"restarting_replicas",
            override.getOrDefault("restarting_replicas", 0),
            "started_replicas",
            override.getOrDefault("started_replicas", 0)
        );
    }
    private static IndexRoutingTable index(String name, ShardAllocation primaryState, ShardAllocation... replicaStates) {
        return index(
            IndexMetadata.builder(name)
                .settings(Settings.builder().put(IndexMetadata.SETTING_VERSION_CREATED, IndexVersion.current()).build())
                .numberOfShards(1)
                .numberOfReplicas(replicaStates.length)
                .build(),
            primaryState,
            replicaStates
        );
    }
    private static IndexRoutingTable index(IndexMetadata indexMetadata, ShardAllocation primaryState, ShardAllocation... replicaStates) {
        var index = indexMetadata.getIndex();
        var shardId = new ShardId(index, 0);
        var builder = IndexRoutingTable.builder(index);
        builder.addShard(createShardRouting(shardId, true, primaryState));
        for (var replicaState : replicaStates) {
            builder.addShard(createShardRouting(shardId, false, replicaState));
        }
        return builder.build();
    }
    private static ShardRouting createShardRouting(ShardId shardId, boolean primary, ShardAllocation allocation) {
        var routing = newUnassigned(
            shardId,