### 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
import org.mockito.stubbing.Answer;
import java.util.ArrayList;
import java.util.Collections;
import java.util.HashMap;
import java.util.List;
import java.util.Map;
import java.util.Optional;
import java.util.Set;
import java.util.UUID;
import java.util.stream.Collectors;
import static java.util.Collections.emptyList;
import static java.util.stream.Collectors.toMap;
import static org.elasticsearch.cluster.metadata.DataStreamTestHelper.createBackingIndex;
import static org.elasticsearch.cluster.metadata.DataStreamTestHelper.newInstance;
import static org.elasticsearch.cluster.metadata.IndexMetadata.INDEX_ROUTING_INCLUDE_GROUP_PREFIX;
import static org.elasticsearch.cluster.metadata.IndexMetadata.INDEX_ROUTING_REQUIRE_GROUP_PREFIX;
import static org.elasticsearch.cluster.metadata.SingleNodeShutdownMetadata.Type.RESTART;
import static org.elasticsearch.cluster.routing.ShardRouting.newUnassigned;
import static org.elasticsearch.cluster.routing.allocation.ShardsAvailabilityHealthIndicatorService.ACTION_CHECK_ALLOCATION_EXPLAIN_API;
import static org.elasticsearch.cluster.routing.allocation.ShardsAvailabilityHealthIndicatorService.ACTION_ENABLE_CLUSTER_ROUTING_ALLOCATION;
import static org.elasticsearch.cluster.routing.allocation.ShardsAvailabilityHealthIndicatorService.ACTION_ENABLE_INDEX_ROUTING_ALLOCATION;
import static org.elasticsearch.cluster.routing.allocation.ShardsAvailabilityHealthIndicatorService.ACTION_ENABLE_TIERS_LOOKUP;
import static org.elasticsearch.cluster.routing.allocation.ShardsAvailabilityHealthIndicatorService.ACTION_INCREASE_NODE_CAPACITY;
import static org.elasticsearch.cluster.routing.allocation.ShardsAvailabilityHealthIndicatorService.ACTION_INCREASE_SHARD_LIMIT_CLUSTER_SETTING;
import static org.elasticsearch.cluster.routing.allocation.ShardsAvailabilityHealthIndicatorService.ACTION_INCREASE_SHARD_LIMIT_CLUSTER_SETTING_LOOKUP;
import static org.elasticsearch.cluster.routing.allocation.ShardsAvailabilityHealthIndicatorService.ACTION_INCREASE_SHARD_LIMIT_INDEX_SETTING;
import static org.elasticsearch.cluster.routing.allocation.ShardsAvailabilityHealthIndicatorService.ACTION_INCREASE_SHARD_LIMIT_INDEX_SETTING_LOOKUP;
import static org.elasticsearch.cluster.routing.allocation.ShardsAvailabilityHealthIndicatorService.ACTION_INCREASE_TIER_CAPACITY_LOOKUP;
import static org.elasticsearch.cluster.routing.allocation.ShardsAvailabilityHealthIndicatorService.ACTION_MIGRATE_TIERS_AWAY_FROM_INCLUDE_DATA;
import static org.elasticsearch.cluster.routing.allocation.ShardsAvailabilityHealthIndicatorService.ACTION_MIGRATE_TIERS_AWAY_FROM_INCLUDE_DATA_LOOKUP;