### 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(ShardsAvailabilityHealthIndicatorService.NAME, equalTo("shards_availability"));
        assertThat(
            ACTION_RESTORE_FROM_SNAPSHOT.getUniqueId(),
            equalTo("elasticsearch:health:shards_availability:diagnosis:restore_from_snapshot")
        );
        assertThat(
            ACTION_CHECK_ALLOCATION_EXPLAIN_API.getUniqueId(),
            equalTo("elasticsearch:health:shards_availability:diagnosis:explain_allocations")
        );
        assertThat(
            DIAGNOSIS_WAIT_FOR_OR_FIX_DELAYED_SHARDS.getUniqueId(),
            equalTo("elasticsearch:health:shards_availability:diagnosis:delayed_shard_allocations")
        );
        assertThat(
            ACTION_ENABLE_INDEX_ROUTING_ALLOCATION.getUniqueId(),
            equalTo("elasticsearch:health:shards_availability:diagnosis:enable_index_allocations")
        );
        assertThat(
            ACTION_ENABLE_CLUSTER_ROUTING_ALLOCATION.getUniqueId(),
            equalTo("elasticsearch:health:shards_availability:diagnosis:enable_cluster_allocations")
        );
        assertThat(
            ACTION_INCREASE_SHARD_LIMIT_CLUSTER_SETTING.getUniqueId(),
            equalTo("elasticsearch:health:shards_availability:diagnosis:increase_shard_limit_cluster_setting")
        );
        assertThat(
            ACTION_INCREASE_SHARD_LIMIT_INDEX_SETTING.getUniqueId(),
            equalTo("elasticsearch:health:shards_availability:diagnosis:increase_shard_limit_index_setting")
        );
        assertThat(