But this gives you an AWS URL, and fetching that in external apps leads to a lot of CORS issues. Isn't any other endpoint available for retrieving the transcript without "leaving" Genesys Cloud (=> relying on AWS resources)?
That is a Genesys resource; the Genesys Cloud services are built entirely in AWS.
Please open a case with Genesys Cloud Care to report the lack of CORS headers as a bug. I doubt the behavior to prohibit all web apps from accessing the transcript files is intentional as the API resource allows CORS. In the meantime, you can implement a web proxy to apply the CORS headers so your web app can access the files.