### 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
frozenCacheService.get()
            );
        });
    }
    @Override
    public Optional<EngineFactory> getEngineFactory(IndexSettings indexSettings) {
        if (indexSettings.getIndexMetadata().isSearchableSnapshot()) {
            final Boolean frozen = indexSettings.getSettings().getAsBoolean("index.frozen", null);
            final boolean useFrozenEngine = indexSettings.getIndexMetadata().isPartialSearchableSnapshot()
                && (frozen == null || frozen.equals(Boolean.TRUE));
            if (useFrozenEngine) {
                return Optional.of(
                    engineConfig -> new FrozenEngine(
                        engineConfig,
                        null,
                        new TranslogStats(),
                        false,
                        indexSettings.getValue(SourceOnlySnapshotRepository.SOURCE_ONLY)
                            ? SourceOnlySnapshotRepository.readerWrapper(engineConfig)
                            : Function.identity(),
                        false,
                        true
                    )
                );
            } else {
                return Optional.of(
                    engineConfig -> new ReadOnlyEngine(
                        engineConfig,
                        null,
                        new TranslogStats(),