I've been adding a Load Balancer to my Cloud Run apps via the Google Cloud Console: Cloud Run => my app => Integrations => Add integration.
This has worked and allowed me to set each app to a custom domain.
For each new app, the right sidebar int he console that appears for "Add integration" shows the existing app integrations. HOWEVER, now the existing integrations are not shown. I went ahead and added an integration for my new app and ALL EXISTING INTEGRATIONS WERE DELETED (including the associated IAP setups for each app)!
Now, when I click Cloud Run => my app => Integrations => Add integration, NONE of the existing integrations are shown, so if I add a new integration, any existing integrations are deleted! I can no longer add integrations as I previously did. If the UI of the GCP console was recently updated, then developers should now at least get a clear warning that Cloud Run app integrations cannot be added as previously done.
To be clear: all of my apps are no longer available at the custom domains that I set up for each. None of my users can use any of my apps, and I don't know how to add all of the Load Balancer integrations (see above for the issue).