It's been my general experience that startups, especially startups built by dev-focussed folk, tend to radically underestimate the impact of people issues (moral, amount of sleep, work/life balance, etc.) on their effectiveness of their business.
In fact I saw almost exactly this story, but with dev rather than support. Going from _perception_ of their being an ever growing list of new features, massive overtime, general feeling of being overwhelmed. To showing that folk were actually getting on top of things with numbers. To some scope adjustment so we could make the hours sane. The team getting happier, and faster at every step.
The other thing is we had focused our entire 5 person dev team on helping support with bug fixes and help text / documentation for common problems. That took a while to kick in and lower the support load, but it's definitely helped!
You can solve a lot of problems through better time management, more efficient processes and improved focus.