_simulate_index should remove private settings

Templates cannot be created with private settings as only Elasticsearch itself may add private settings. But `/_index_template/_simulate_index` can currently return private settings. Since templates themselves cannot contains private settings, it does not make sense for the templates returned by `_simulate_index` to contain private settings. Also, since the results of `_simulate_index` are frequently used to create a template or index, a private settings would result in a failure to create the index or template.

The solutions is likely to just remove any private settings created by settings providers, here: https://github.com/elastic/elasticsearch/blob/bce397f40507c262d8b47c885dea02fa46ac0a93/server/src/main/java/org/elasticsearch/action/admin/indices/template/post/TransportSimulateIndexTemplateAction.java#L294

This issue showed up in a failed Kibana test: https://github.com/elastic/kibana/issues/235296 . This tests attempts to create an index using the results of a `_simulate_index` call. This cannot be fixed within Kibana, since it has no way of determining if a given settings is private or not.

Example showing that `_simulate_index` returns private settings:
```

# Create logsdb index with host.name
curl -X PUT "localhost:9200/_index_template/temp1?pretty" -H 'Content-Type: application/json' -d'
{
"index_patterns": ["my-logsdb-datastream-*"],
"data_stream": {},
"template": {
"settings": {
"index.mode": "logsdb",
"number_of_shards": 1
},
"mappings": {
"properties": {
"timestamp": {
"type": "date",
"format": "strict_date_optional_time||epoch_millis"
},
"message": {
"type": "text"
},
"host.name": { "type": "keyword"}
}
}
},
"priority": 200
}'

curl -XPUT localhost:9200/_data_stream/my-logsdb-datastream-5

# sort_on_host_name (which is private) was set
curl localhost:9200/.ds-my-logsdb-datastream-5-2025.10.22-000001/_settings
>>>
...
"logsdb": {
"sort_on_host_name": "true"
},

# Simulate_index also contains private setting
curl -XPOST "localhost:9200/_index_template/_simulate_index/my-logsdb-datastream-5" | jq
>>>
"template": {
"settings": {
"index": {
"mode": "logsdb",
"routing": {
"allocation": {
"include": {
"_tier_preference": "data_hot"
}
}
},
"number_of_shards": "1",
"logsdb": {
"sort_on_host_name": "true"
}
}
},
...

```

Start in server/src/main/java/org/elasticsearch/action/admin/indices/template/post/TransportSimulateIndexTemplateAction.java around line 294, where settings providers contribute to the simulated template. Reproduce the example with _index_template/_simulate_index and verify that the response no longer includes private settings and can be used to create a template or index.

Written by the indexing model from the issue text.