### 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
}
        public void addPrimary(ShardRouting routing, ClusterState state, NodesShutdownMetadata shutdowns, boolean verbose) {
            primaries.increment(routing, state, shutdowns, verbose);
        }
        public void addReplica(ShardRouting routing, ClusterState state, NodesShutdownMetadata shutdowns, boolean verbose) {
            replicas.increment(routing, state, shutdowns, verbose);
        }
        public HealthStatus getStatus() {
            if (primaries.available == false) {
                return RED;
            } else if (replicas.available == false) {
                return YELLOW;
            } else {
                return GREEN;
            }
        }
        public String getSymptom() {
            var builder = new StringBuilder("This cluster has ");
            if (primaries.unassigned > 0
                || primaries.unassigned_new > 0
                || primaries.unassigned_restarting > 0
                || replicas.unassigned > 0
                || replicas.unassigned_new > 0
                || replicas.unassigned_restarting > 0
                || primaries.initializing > 0
                || replicas.initializing > 0) {
                builder.append(
                    Stream.of(
                        createMessage(primaries.unassigned, "unavailable primary shard", "unavailable primary shards"),
                        createMessage(primaries.unassigned_new, "creating primary shard", "creating primary shards"),