Hi
Thank you for posting query in Microsoft Q&A Platform.
yes you are right at this point of time, we have portal or SSMS or PowerShell as possible way for doing this. APIs are not available. Please check below link.
https://learn.microsoft.com/en-us/azure/analysis-services/analysis-services-backup
It's good feedback. I would encourage you to have log feedback item for this using below link. Internal team closly check feedbacks and consider them for future.
https://feedback.azure.com/d365community/forum/d92323a0-0525-ec11-b6e6-000d3a4f0858
Hope this helps. Please let me know if any further queries.
Please consider hitting Accept Answer
button. Accepted answers help community as well. Thank you.