### 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
index("index-1", new ShardAllocation(randomNodeId(), AVAILABLE), new ShardAllocation(randomNodeId(), UNAVAILABLE)),
                index("index-2", new ShardAllocation(randomNodeId(), AVAILABLE), new ShardAllocation(randomNodeId(), UNAVAILABLE))
            ),
            List.of(),
            List.of()
        );
        var service = createShardsAvailabilityIndicatorService(clusterState);
        HealthIndicatorResult result = service.calculate(true, HealthInfo.EMPTY_HEALTH_INFO);
        
        assertThat(
            result.impacts().get(0),
            equalTo(
                new HealthIndicatorImpact(
                    NAME,
                    ShardsAvailabilityHealthIndicatorService.REPLICA_UNASSIGNED_IMPACT_ID,
                    2,
                    "Searches might be slower than usual. Fewer redundant copies of the data exist on 3 indices [index-2, "
                        + "index-1, index-3].",
                    List.of(ImpactArea.SEARCH)
                )
            )
        );
    }
    public void testShouldBeGreenWhenThereAreRestartingReplicas() {
        var clusterState = createClusterStateWith(
            List.of(
                index(
                    "restarting-index",
                    new ShardAllocation(randomNodeId(), AVAILABLE),
                    new ShardAllocation("node-0", RESTARTING, System.nanoTime())