### 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
if (shutdown == null) {
            return false;
        }
        var now = System.nanoTime();
        var restartingAllocationDelayExpiration = info.getUnassignedTimeInNanos() + shutdown.getAllocationDelay().nanos();
        return now - restartingAllocationDelayExpiration <= 0;
    }
    private static boolean isUnassignedDueToNewInitialization(ShardRouting routing, ClusterState state) {
        if (routing.active()) {
            return false;
        }
        
        
        ShardRouting primary = routing.primary() ? routing : state.routingTable().shardRoutingTable(routing.shardId()).primaryShard();
        return primary.active() == false && getInactivePrimaryHealth(primary) == ClusterHealthStatus.YELLOW;
    }
    /**
     * Generate a list of diagnoses that'll contain the instructions for a user to take to allow this shard to be assigned.
     * @param shardRouting An unassigned shard routing
     * @param state State of the cluster
     * @return A list of diagnoses for the provided unassigned shard
     */
    List<Diagnosis.Definition> diagnoseUnassignedShardRouting(ShardRouting shardRouting, ClusterState state) {
        List<Diagnosis.Definition> diagnosisDefs = new ArrayList<>();
        LOGGER.trace("Diagnosing unassigned shard [{}] due to reason [{}]", shardRouting.shardId(), shardRouting.unassignedInfo());
        switch (shardRouting.unassignedInfo().getLastAllocationStatus()) {
            case NO_VALID_SHARD_COPY -> diagnosisDefs.add(ACTION_RESTORE_FROM_SNAPSHOT);
            case NO_ATTEMPT -> {
                if (shardRouting.unassignedInfo().isDelayed()) {
                    diagnosisDefs.add(DIAGNOSIS_WAIT_FOR_OR_FIX_DELAYED_SHARDS);