I had misunderstood. I thought you had meant the character login screen at first, rather than the actual game login screen itself.

I notice you haven't mentioned anything about firewall settings? Is it possible ffxiv is being blocked for some reason? I'd def double check both the windows firewall settings and any other antivirus/firewall software you may have. I've had this issue myself a few weeks after I first started playing, I think this is what it was.