The console dump was super useful, thank you for that!
I think it might be because you have third-party cookies disabled [1], and the Google Auth client uses an iframe with cookies to do its thing. I replicated it myself by disabling third-party cookies and the results were similar to what you described.
I'll look into what I can do about this on my end tomorrow. Thank you again for your time!
Interesting. I log into a handful of other sites using Google OAuth and have never had a problem. Let me know if you get it figured out, I would still like to check out your service.
If you would still like to give it another shot I would appreciate it :)