### 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
public static final Diagnosis.Definition ACTION_ENABLE_INDEX_ROUTING_ALLOCATION = new Diagnosis.Definition(
        NAME,
        "enable_index_allocations",
        "Elasticsearch isn't allowed to allocate some shards from these indices because allocation for those shards has been disabled at "
            + "the index level.",
        "Check that the ["
            + INDEX_ROUTING_ALLOCATION_ENABLE_SETTING.getKey()
            + "] index settings are set to ["
            + EnableAllocationDecider.Allocation.ALL.toString().toLowerCase(Locale.getDefault())
            + "].",
        ENABLE_INDEX_ALLOCATION_GUIDE
    );
    public static final String ENABLE_CLUSTER_ALLOCATION_ACTION_GUIDE = "https://ela.st/fix-cluster-allocation";
    public static final Diagnosis.Definition ACTION_ENABLE_CLUSTER_ROUTING_ALLOCATION = new Diagnosis.Definition(
        NAME,
        "enable_cluster_allocations",
        "Elasticsearch isn't allowed to allocate some shards from these indices because allocation for those shards has been disabled at "
            + "the cluster level.",
        "Check that the ["
            + EnableAllocationDecider.CLUSTER_ROUTING_ALLOCATION_ENABLE_SETTING.getKey()
            + "] cluster setting is set to ["
            + EnableAllocationDecider.Allocation.ALL.toString().toLowerCase(Locale.getDefault())
            + "].",
        ENABLE_CLUSTER_ALLOCATION_ACTION_GUIDE
    );
    public static final String ENABLE_TIER_ACTION_GUIDE = "https://ela.st/enable-tier";
    public static final Map<String, Diagnosis.Definition> ACTION_ENABLE_TIERS_LOOKUP = DataTier.ALL_DATA_TIERS.stream()
        .collect(
            Collectors.toUnmodifiableMap(
                tier -> tier,