### 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
ShardsAvailabilityHealthIndicatorService.PRIMARY_UNASSIGNED_IMPACT_ID,
                                1,
                                "Cannot add data to 5 indices [red-index1, red-index2, red-index3, red-index4, red-index5]. 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-index1", "red-index2")))
                            )
                        )
                    )
                )
            );
        }
        {
            
            assertThat(
                service.calculate(true, 2_000, HealthInfo.EMPTY_HEALTH_INFO).diagnosisList(),
                equalTo(
                    List.of(
                        new Diagnosis(
                            ACTION_CHECK_ALLOCATION_EXPLAIN_API,
                            List.of(
                                new Diagnosis.Resource(INDEX, List.of("red-index1", "red-index2", "red-index3", "red-index4", "red-index5"))
                            )
                        )
                    )