### 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
primary,
            getSource(primary, allocation.state),
            new UnassignedInfo(UnassignedInfo.Reason.INDEX_CREATED, null),
            ShardRouting.Role.DEFAULT
        );
        if (allocation.state == CREATING) {
            return routing;
        }
        routing = routing.initialize(allocation.nodeId, null, 0);
        if (allocation.state == INITIALIZING) {
            return routing;
        }
        routing = routing.moveToStarted(ShardRouting.UNAVAILABLE_EXPECTED_SHARD_SIZE);
        if (allocation.state == AVAILABLE) {
            return routing;
        }
        if (allocation.state == UNAVAILABLE) {
            return routing.moveToUnassigned(Optional.ofNullable(allocation.unassignedInfo).orElse(randomFrom(nodeLeft(), decidersNo())));
        }
        if (allocation.state == RESTARTING) {
            return routing.moveToUnassigned(
                new UnassignedInfo(
                    UnassignedInfo.Reason.NODE_RESTARTING,
                    null,
                    null,
                    -1,
                    allocation.unassignedTimeNanos != null ? allocation.unassignedTimeNanos : 0,
                    0,
                    false,
                    UnassignedInfo.AllocationStatus.DELAYED_ALLOCATION,