### 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
unassigned_restarting++;
                    } else {
                        unassigned++;
                        if (verbose) {
                            diagnoseUnassignedShardRouting(routing, state).forEach(
                                definition -> addDefinition(definition, routing.getIndexName())
                            );
                        }
                    }
                }
                case INITIALIZING -> {
                    initializing++;
                    if (verbose) {
                        addDefinition(DIAGNOSIS_WAIT_FOR_INITIALIZATION, routing.getIndexName());
                    }
                }
                case STARTED -> started++;
                case RELOCATING -> relocating++;
            }
        }
        private void addDefinition(Diagnosis.Definition diagnosisDefinition, String indexName) {
            diagnosisDefinitions.computeIfAbsent(diagnosisDefinition, (k) -> new HashSet<>()).add(indexName);
        }
    }
    private static boolean isUnassignedDueToTimelyRestart(ShardRouting routing, NodesShutdownMetadata shutdowns) {
        var info = routing.unassignedInfo();
        if (info == null || info.getReason() != UnassignedInfo.Reason.NODE_RESTARTING) {
            return false;
        }
        var shutdown = shutdowns.get(info.getLastAllocatedNodeId(), SingleNodeShutdownMetadata.Type.RESTART);