### 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
ACTION_MIGRATE_TIERS_AWAY_FROM_REQUIRE_DATA.getUniqueId(),
            equalTo("elasticsearch:health:shards_availability:diagnosis:migrate_data_tiers_require_data")
        );
        assertThat(
            ACTION_MIGRATE_TIERS_AWAY_FROM_INCLUDE_DATA.getUniqueId(),
            equalTo("elasticsearch:health:shards_availability:diagnosis:migrate_data_tiers_include_data")
        );
        assertThat(
            ACTION_INCREASE_NODE_CAPACITY.getUniqueId(),
            equalTo("elasticsearch:health:shards_availability:diagnosis:increase_node_capacity_for_allocations")
        );
        assertThat(
            DIAGNOSIS_WAIT_FOR_INITIALIZATION.getUniqueId(),
            equalTo("elasticsearch:health:shards_availability:diagnosis:initializing_shards")
        );
        for (String tier : List.of("data_content", "data_hot", "data_warm", "data_cold", "data_frozen")) {
            assertThat(
                ACTION_ENABLE_TIERS_LOOKUP.get(tier).getUniqueId(),
                equalTo("elasticsearch:health:shards_availability:diagnosis:enable_data_tiers:tier:" + tier)
            );
            assertThat(
                ACTION_INCREASE_SHARD_LIMIT_INDEX_SETTING_LOOKUP.get(tier).getUniqueId(),
                equalTo("elasticsearch:health:shards_availability:diagnosis:increase_shard_limit_index_setting:tier:" + tier)
            );
            assertThat(
                ACTION_INCREASE_SHARD_LIMIT_CLUSTER_SETTING_LOOKUP.get(tier).getUniqueId(),
                equalTo("elasticsearch:health:shards_availability:diagnosis:increase_shard_limit_cluster_setting:tier:" + tier)
            );
            assertThat(
                ACTION_MIGRATE_TIERS_AWAY_FROM_REQUIRE_DATA_LOOKUP.get(tier).getUniqueId(),