I very recently installed the Safari extension, and I'm surprised there isn't more 'traction' on this annoying issue.
The the inability to _not_ launch the AdGuard settings window every time you launch Safari is a major nuisance - and completely irrational. Why should we need to see the extension settings every time we launch the the browser app? We can already easily tweak those whenever we need to - there is no reason to have a huge, pointless window obscuring the app on initial launch.
Also you can try this for a temporary fix that only works if you initially installed from the App Store:
1: Use a tool like AppCleaner to remove AdGuard and everything associated with the app for a proper fresh start.
2: Download the non-app store release version: https://agrd.io/safari_release.
3: Install it, let it start on start up and say yes when prompted about doing this automatically next time.
4: Reboot and enjoy a pop-up free AdGuard start up, all you see is the icon quickly pop in and out of the dock.