### 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
}
        }
        return diagnosisDefs;
    }
    /**
     * Generates a list of diagnoses for common problems that keep a shard from allocating to nodes in a data tier.
     * @param indexMetadata Index metadata for the shard being diagnosed.
     * @param nodeAllocationResults allocation decision results for all nodes in the cluster.
     * @param clusterState the current cluster state.
     * @return A list of diagnoses for the provided unassigned shard
     */
    List<Diagnosis.Definition> checkDataTierRelatedIssues(
        IndexMetadata indexMetadata,
        List<NodeAllocationResult> nodeAllocationResults,
        ClusterState clusterState
    ) {
        List<Diagnosis.Definition> diagnosisDefs = new ArrayList<>();
        if (indexMetadata.getTierPreference().size() > 0) {
            List<NodeAllocationResult> dataTierAllocationResults = nodeAllocationResults.stream()
                .filter(hasDeciderResult(DATA_TIER_ALLOCATION_DECIDER_NAME, Decision.Type.YES))
                .toList();
            if (dataTierAllocationResults.isEmpty()) {
                
                for (String tier : indexMetadata.getTierPreference()) {
                    Optional.ofNullable(ACTION_ENABLE_TIERS_LOOKUP.get(tier)).ifPresent(diagnosisDefs::add);
                }
            } else {
                
                Set<DiscoveryNode> dataTierNodes = dataTierAllocationResults.stream()
                    .map(NodeAllocationResult::getNode)