Edge Job ID - is there any way to get access to this outside of the Edge Jobs Beta?

We have had some profiling jobs completing with errors for some time. I went to get the Job ID to pull the connector logs, and realized the Activity did not have a Job ID associated with these results. I then went to the Edge Jobs beta section, and that Job ID is the correct one for the connector.

The issue there is that it does not list the specific asset it is being run against, and only lists success or failure. I figured I could pull the Job IDs from the REST Core API, but that is not the “Job Id” it is the Id of that item that references the job in the API.

I had thought awhile back that there was some mention of another API call that could pull the actual Edge Job ID? But I cannot seem to find it in the community, or in the REST API documentation. Or is the only way to find these to note the completion time in Activities and find the related item in the Edge Job Beta list?