Thanks for bringing this issue to our attention. As far as I know, there isn't a difference in setting up the auto-renewal process for App Service Certificate vs. App Service Managed Certificate. Can you please share with us what steps you followed to automatically renew your managed cert?
Based on this doc, these are the steps to auto-renew managed certs:
To turn on automatic renewal of your certificate at any time, select the certificate in the App Service Certificates page, then click Auto Renew Settings in the left navigation.
Select On and click Save. Certificates can start automatically renewing 60 days before expiration if you have the automatic renewal turned on.
Renew App Service certificate automatically
Once the renew operation is complete, click Sync. The sync operation automatically updates the hostname bindings for the certificate in App Service without causing any downtime to your apps.
Hope that helps. Please let us know if you have further questions.
Thanks,
Grace