Three interrelated issues here:

- [ ] `TransportBroadcastByNodeAction` derivatives are typically executed via `NodeClient` which bypasses the `TransportService` and therefore does not respect (or even know about) the `executor` parameter which would fork. This means that the coordination work, including grouping all the shards by node, happens on the calling thread. When executed from the REST layer, that's a transport worker. This is #97916, but until that's fixed we need a workaround.

- [ ] The node-level responses are deserialised and processed on the receiving transport worker too.

- [ ] Several of these actions use `SAME` for their executor, bypassing the forking that _does_ exist today.

Relates #77466 
server/src/main/java/org/elasticsearch/action/admin/indices/stats/ShardStats.java
static final String ROUTING = "routing";
        static final String STATE = "state";
        static final String STATE_PATH = "state_path";
        static final String DATA_PATH = "data_path";
        static final String IS_CUSTOM_DATA_PATH = "is_custom_data_path";
        static final String SHARD_PATH = "shard_path";
        static final String PRIMARY = "primary";
        static final String NODE = "node";
        static final String RELOCATING_NODE = "relocating_node";
        static final String SEARCH_IDLE = "search_idle";
        static final String SEARCH_IDLE_TIME = "search_idle_time";
    }
}