### 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
for (int i = 0; i < indexShardRouting.size(); i++) {
                IndexShardRoutingTable shardRouting = indexShardRouting.shard(i);
                status.addPrimary(shardRouting.primaryShard(), state, shutdown, verbose);
                for (ShardRouting replicaShard : shardRouting.replicaShards()) {
                    status.addReplica(replicaShard, state, shutdown, verbose);
                }
            }
        }
        return createIndicator(
            status.getStatus(),
            status.getSymptom(),
            status.getDetails(verbose),
            status.getImpacts(),
            status.getDiagnosis(verbose, maxAffectedResourcesCount)
        );
    }
    
    public static final String PRIMARY_UNASSIGNED_IMPACT_ID = "primary_unassigned";
    public static final String REPLICA_UNASSIGNED_IMPACT_ID = "replica_unassigned";
    public static final String RESTORE_FROM_SNAPSHOT_ACTION_GUIDE = "https://ela.st/restore-snapshot";
    public static final Diagnosis.Definition ACTION_RESTORE_FROM_SNAPSHOT = new Diagnosis.Definition(
        NAME,
        "restore_from_snapshot",
        "Elasticsearch isn't allowed to allocate some shards because there are no copies of the shards in the cluster. Elasticsearch will "
            + "allocate these shards when nodes holding good copies of the data join the cluster.",
        "If no such node is available, restore these indices from a recent snapshot.",
        RESTORE_FROM_SNAPSHOT_ACTION_GUIDE
    );
    public static final String DIAGNOSE_SHARDS_ACTION_GUIDE = "https://ela.st/diagnose-shards";
    public static final Diagnosis.Definition ACTION_CHECK_ALLOCATION_EXPLAIN_API = new Diagnosis.Definition(