### 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
),
            ClusterState.EMPTY_STATE
        );
        assertThat(actions, hasSize(1));
        assertThat(actions, contains(ACTION_INCREASE_NODE_CAPACITY));
    }
    public void testLimitNumberOfAffectedResources() {
        var clusterState = createClusterStateWith(
            List.of(
                index("red-index1", new ShardAllocation(randomNodeId(), UNAVAILABLE)),
                index("red-index2", new ShardAllocation(randomNodeId(), UNAVAILABLE)),
                index("red-index3", new ShardAllocation(randomNodeId(), UNAVAILABLE)),
                index("red-index4", new ShardAllocation(randomNodeId(), UNAVAILABLE)),
                index("red-index5", new ShardAllocation(randomNodeId(), UNAVAILABLE))
            ),
            List.of()
        );
        var service = createShardsAvailabilityIndicatorService(clusterState);
        {
            
            assertThat(
                service.calculate(true, 2, HealthInfo.EMPTY_HEALTH_INFO),
                equalTo(
                    createExpectedResult(
                        RED,
                        "This cluster has 5 unavailable primary shards.",
                        Map.of("unassigned_primaries", 5),
                        List.of(
                            new HealthIndicatorImpact(
                                NAME,