I disagree, for the past 4 years, I've worked in remote teams that were across different timezones and different countries. There are days were half the team is offline because it's a holiday in their country, and it doesn't really impact our team. We know that some people will have a day off, and that's fine, we don't bother them, and nobody expects them to be online.
Same with timezone differences, it's generally understood we work 9-6. If it's after 6PM in someone's time, we don't bother them, because we expect them to do the same. Does it take some more planning? Sure it does, if I want to talk to someone on the same day, I just use Skype to find a time that works for the both of us. Otherwise I schedule a meeting for the next day.
> Then they realize it's really important and need to chat right now.
Having a "really important" issue where you need to chat with someone right now is usually just a matter of perception. Usually those issues can wait a couple of hours, or even a couple of days.
All in all, working across different timezones and countries works perfectly fine, I have plenty of time during the day to schedule calls with my colleagues.