### 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
"Elasticsearch isn't allowed to allocate some shards from these indices to any nodes in the ["
                        + tier
                        + "] data tier because the indices are configured with allocation filter rules that are incompatible with the "
                        + "nodes in this tier.",
                    "Remove ["
                        + INDEX_ROUTING_INCLUDE_GROUP_PREFIX
                        + ".data] from the index settings or try migrating to data tiers by first stopping ILM [POST /_ilm/stop] and then "
                        + "using the data tier migration action [POST /_ilm/migrate_to_data_tiers]. Finally, restart ILM "
                        + "[POST /_ilm/start].",
                    MIGRATE_TO_TIERS_ACTION_GUIDE
                )
            )
        );
    public static final String TIER_CAPACITY_ACTION_GUIDE = "https://ela.st/tier-capacity";
    public static final Diagnosis.Definition ACTION_INCREASE_NODE_CAPACITY = new Diagnosis.Definition(
        NAME,
        "increase_node_capacity_for_allocations",
        "Elasticsearch isn't allowed to allocate some shards from these indices because there are not enough nodes in the cluster to "
            + "allocate each shard copy on a different node.",
        "Increase the number of nodes in the cluster or decrease the number of replica shards in the affected indices.",
        TIER_CAPACITY_ACTION_GUIDE
    );
    public static final Map<String, Diagnosis.Definition> ACTION_INCREASE_TIER_CAPACITY_LOOKUP = DataTier.ALL_DATA_TIERS.stream()
        .collect(
            Collectors.toUnmodifiableMap(
                tier -> tier,
                tier -> new Diagnosis.Definition(
                    NAME,
                    "increase_tier_capacity_for_allocations:tier:" + tier,
                    "Elasticsearch isn't allowed to allocate some shards from these indices to any of the nodes in the desired data tier "