I should point out that the documentation for that API indicates that it can be used for Webinars as well:
First, have you confirmed that the latest instance of the Webinar ID has a recording? If there have been multiple occurrences of the Webinar it’s possible that the latest doesn’t have a recording. You can try using the UUID of the instance you want to target to make sure that you are referencing the correct instance.
Let me know if that helps. If not, Please send an email to email@example.com with a link to this thread. In that email, please include the request that you’re making including the Webinar ID.