### 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
+ "because there are not enough nodes in the ["
                        + tier
                        + "] tier to allocate each shard copy on a different node.",
                    "Increase the number of nodes in this tier or decrease the number of replica shards in the affected indices.",
                    TIER_CAPACITY_ACTION_GUIDE
                )
            )
        );
    private class ShardAllocationCounts {
        private boolean available = true; 
        private int unassigned = 0;
        private int unassigned_new = 0;
        private int unassigned_restarting = 0;
        private int initializing = 0;
        private int started = 0;
        private int relocating = 0;
        private final Set<String> indicesWithUnavailableShards = new HashSet<>();
        private final Map<Diagnosis.Definition, Set<String>> diagnosisDefinitions = new HashMap<>();
        public void increment(ShardRouting routing, ClusterState state, NodesShutdownMetadata shutdowns, boolean verbose) {
            boolean isNew = isUnassignedDueToNewInitialization(routing, state);
            boolean isRestarting = isUnassignedDueToTimelyRestart(routing, shutdowns);
            available &= routing.active() || isRestarting || isNew;
            if ((routing.active() || isRestarting || isNew) == false) {
                indicesWithUnavailableShards.add(routing.getIndexName());
            }
            switch (routing.state()) {
                case UNASSIGNED -> {
                    if (isNew) {
                        unassigned_new++;
                    } else if (isRestarting) {