### 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_
x-pack/plugin/searchable-snapshots/src/main/java/org/elasticsearch/xpack/searchablesnapshots/SearchableSnapshots.java
builder.endObject();
                    }
                    {
                        builder.startObject("properties");
                        {
                            builder.startObject("type");
                            builder.field("type", "keyword");
                            builder.endObject();
                        }
                        {
                            builder.startObject("creation_time");
                            builder.field("type", "date");
                            builder.field("format", "epoch_millis");
                            builder.endObject();
                        }
                        {
                            builder.startObject("version");
                            builder.field("type", "integer");
                            builder.endObject();
                        }
                        {
                            builder.startObject("repository");
                            builder.field("type", "keyword");
                            builder.endObject();
                        }
                        {
                            builder.startObject("blob");
                            builder.field("type", "object");
                            {
                                builder.startObject("properties");