In my opinion, the simplest solution is to use GitHub notifications directly.
On the repository page you can enable Watch → Custom → Releases.
As soon as a new version is published, you’ll get an email notification with a direct link to the release page. From there it’s literally two clicks — one to open the release, another to download the AppImage.
you get notified immediately when a new version is out
the download is just two clicks away
no need for custom scripts or dealing with the API
For me this feels like the cleanest and least complicated approach.
Thanks for your suggestions
I remebered, that i can just add a Github-link in Gear lever instead of a static one and it will automatically update if there is a new one available, so that solves all my issues.