### 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
1,
                "Cannot add data to 1 index [red-index]. Searches might return incomplete results.",
                List.of(ImpactArea.INGEST, ImpactArea.SEARCH)
            )
        );
        
        assertThat(
            result.impacts().get(1),
            equalTo(
                new HealthIndicatorImpact(
                    NAME,
                    ShardsAvailabilityHealthIndicatorService.REPLICA_UNASSIGNED_IMPACT_ID,
                    2,
                    "Searches might be slower than usual. Fewer redundant copies of the data exist on 2 indices [yellow-index-2, "
                        + "yellow-index-1].",
                    List.of(ImpactArea.SEARCH)
                )
            )
        );
    }
    public void testSortByIndexPriority() {
        var lowPriority = randomIntBetween(1, 5);
        var highPriority = randomIntBetween(6, 20);
        List<IndexMetadata> indexMetadataList = createIndexMetadataForIndexNameToPriorityMap(
            Map.of("index-3", lowPriority, "index-1", lowPriority, "index-2", highPriority)
        );
        var clusterState = createClusterStateWith(
            indexMetadataList,
            List.of(
                index("index-3", new ShardAllocation(randomNodeId(), AVAILABLE), new ShardAllocation(randomNodeId(), UNAVAILABLE)),