How to Ship/Parse JSON logs (from Kibana) to Elasticsearch?

It sounds like Filebeat was not configured to parse the JSON logs generated by Kibana (see config from module). The ingest pipeline from the Kibana filebeat module does not do any JSON parsing and expects it to already be parsed. Because of this there should be an error.message field added to each event by the ingest pipeline. Do you see that error? Are you sure the data is being passed to the pipeline?

You can see stats from the ingest pipelines by querying GET _nodes/stats/ingest and looking for kibana-module-copy. It will tell you the number of events and number of failures.