### 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
)
            );
        }
        {
            
            assertThat(
                service.calculate(true, 0, HealthInfo.EMPTY_HEALTH_INFO).diagnosisList(),
                equalTo(List.of(new Diagnosis(ACTION_CHECK_ALLOCATION_EXPLAIN_API, List.of(new Diagnosis.Resource(INDEX, List.of())))))
            );
        }
    }
    /**
     * Creates the {@link SystemIndices} with one standalone system index and a system data stream
     */
    private SystemIndices getSystemIndices(
        String featureDataStreamName,
        String systemDataStreamPattern,
        String standaloneSystemIndexPattern
    ) {
        return new SystemIndices(
            List.of(
                new SystemIndices.Feature(
                    "feature-with-system-index",
                    "testing",
                    List.of(SystemIndexDescriptorUtils.createUnmanaged(standaloneSystemIndexPattern, "feature with index"))
                ),
                new SystemIndices.Feature(
                    "feature-with-system-data-stream",
                    "feature with data stream",
                    List.of(),