Things to try:
1. Please check fs-uae-launcher.log.txt and send logs if there is something there (
https://fs-uae.net/log-files-needed).
2. Try running from a terminal, something like this:
/Applications/"FS-UAE Launcher.app"/Contents/MacOS/fs-uae-launcher
- does the program start then? or if not, are any error messages printed to the terminal?