### 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
} else {
                    diagnosisDefs.addAll(explainAllocationsAndDiagnoseDeciders(shardRouting, state));
                }
            }
            case DECIDERS_NO -> diagnosisDefs.addAll(explainAllocationsAndDiagnoseDeciders(shardRouting, state));
            case DELAYED_ALLOCATION -> diagnosisDefs.add(DIAGNOSIS_WAIT_FOR_OR_FIX_DELAYED_SHARDS);
        }
        if (diagnosisDefs.isEmpty()) {
            diagnosisDefs.add(ACTION_CHECK_ALLOCATION_EXPLAIN_API);
        }
        return diagnosisDefs;
    }
    /**
     * For a shard that is unassigned due to a DECIDERS_NO result, this will explain the allocation and attempt to generate
     * a list of diagnoses that should allow the shard to be assigned.
     * @param shardRouting The shard routing that is unassigned with a last status of DECIDERS_NO
     * @param state Current cluster state
     * @return A list of diagnoses for the provided unassigned shard
     */
    private List<Diagnosis.Definition> explainAllocationsAndDiagnoseDeciders(ShardRouting shardRouting, ClusterState state) {
        LOGGER.trace("Executing allocation explain on shard [{}]", shardRouting.shardId());
        RoutingAllocation allocation = new RoutingAllocation(
            allocationService.getAllocationDeciders(),
            state,
            ClusterInfo.EMPTY,
            SnapshotShardSizeInfo.EMPTY,
            System.nanoTime()
        );
        allocation.setDebugMode(RoutingAllocation.DebugMode.ON);
        ShardAllocationDecision shardAllocationDecision = allocationService.explainShardAllocation(shardRouting, allocation);