I exceedingly find that the addons I maintain break in some edge cases and feel like if I was logging the addon error in some centralized way I'd be able to determine and fix these issues.
Anybody uses any services for remote error logging in your addons? I'd love to hear some ideas.
I use https://sentry.io/ for logging errors in the Vue part of my extension. It is really easy to integrate and very helpful for the most important part of the extension.
In my case, I'm not tried to integrate it into the background script, but I hope it is possible too.
I have seen some extensions using Mixpanel to log events on extensions. So, in your case the error message can be events. That should work. Be careful not to log private data though when you do it.
I personally have not used Mixpanel. Hope this helps.