"It shouldn't be hard to get some shared calendar services running on an extra box somewhere..."
This is hilarious and naive.
Believe me, setting up any kind of shared calendar in an large enterprise environment (read: hospital) today IS hard. You can't just put extra boxes "somewhere" as you need them. That's not how you build an IT infrastructure. You have to think globally. Think about maintenance, system administration, network access, monitoring, security, data backup, software upgrades, etc. And you can bet most "heads-of-something" will want to access those calendars with any device or software they prefer : "Hey I can't sync my cal with Outlook / Evolution / my iPhone / my Windows 7 phone / my Android one / etc. and I don't care why. Just get it working."
Basically, if there's anything you can do at home when toying with your computer, network or iPad, you probably can't and SHOULDN'T do it in a corporate network. And there are *many* very good reasons to that.