### 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
createMessage(replicas.unassigned_new, "creating replica shard", "creating replica shards"),
                        createMessage(primaries.unassigned_restarting, "restarting primary shard", "restarting primary shards"),
                        createMessage(replicas.unassigned, "unavailable replica shard", "unavailable replica shards"),
                        createMessage(primaries.initializing, "initializing primary shard", "initializing primary shards"),
                        createMessage(replicas.initializing, "initializing replica shard", "initializing replica shards"),
                        createMessage(replicas.unassigned_restarting, "restarting replica shard", "restarting replica shards")
                    ).flatMap(Function.identity()).collect(joining(", "))
                ).append(".");
            } else {
                builder.append("all shards available.");
            }
            return builder.toString();
        }
        private static Stream<String> createMessage(int count, String singular, String plural) {
            return switch (count) {
                case 0 -> Stream.empty();
                case 1 -> Stream.of("1 " + singular);
                default -> Stream.of(count + " " + plural);
            };
        }
        public HealthIndicatorDetails getDetails(boolean verbose) {
            if (verbose) {
                return new SimpleHealthIndicatorDetails(
                    Map.of(
                        "unassigned_primaries",
                        primaries.unassigned,
                        "initializing_primaries",
                        primaries.initializing,
                        "creating_primaries",
                        primaries.unassigned_new,