### 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/test/java/org/elasticsearch/cluster/routing/allocation/ShardsAvailabilityHealthIndicatorServiceTests.java
RED,
                    "This cluster has 1 unavailable primary shard, 1 unavailable replica shard.",
                    Map.of("unassigned_primaries", 1, "unassigned_replicas", 1),
                    List.of(
                        new HealthIndicatorImpact(
                            NAME,
                            ShardsAvailabilityHealthIndicatorService.PRIMARY_UNASSIGNED_IMPACT_ID,
                            1,
                            "Cannot add data to 1 index [red-index]. Searches might return incomplete results.",
                            List.of(ImpactArea.INGEST, ImpactArea.SEARCH)
                        )
                    ),
                    List.of(
                        new Diagnosis(ACTION_CHECK_ALLOCATION_EXPLAIN_API, List.of(new Diagnosis.Resource(INDEX, List.of("red-index"))))
                    )
                )
            )
        );
    }
    public void testShouldBeRedWhenThereAreUnassignedPrimariesAndNoReplicas() {
        var clusterState = createClusterStateWith(List.of(index("red-index", new ShardAllocation(randomNodeId(), UNAVAILABLE))), List.of());
        var service = createShardsAvailabilityIndicatorService(clusterState);
        assertThat(
            service.calculate(true, HealthInfo.EMPTY_HEALTH_INFO),
            equalTo(
                createExpectedResult(
                    RED,
                    "This cluster has 1 unavailable primary shard.",
                    Map.of("unassigned_primaries", 1),
                    List.of(