Here's a little secret of System Administration: Much of being a good admin is your skill and knowledge; but more of it is your wisdom and caution. You don't have enough people to support convenient one-offs all over the place. Keeping the environment manageable (and not just by you) is most of the battle.
Your fellow Docs want an electronic calendar. You don't say that the current system isn't working, just that they would like the more convenient electronic function. Perfectly reasonable. But, it's not reasonable to do an end run around IT. I know that we're sometimes slow to get something done. I know that we can seem very bureaucratic. But you have to understand why and help us to help you. We're slow to get things done because we have way more work to do than we can get done, so we have to prioritize. Your little calendar is not a high priority. If you want it worked on promptly, make some waves in the budget process so IT can get more staff.
As for the bureaucracy, I really think people don't understand that much, sometimes even most of the regulatory burden falls on IT. We do the rights and access work to the data. We produce the reports. We have to write the polices and procedures. We face the auditors. We burn when something goes wrong. None of this is our core function. We'd rather be coding or installing your calendar. These regulations are written with the operational staff in mind, but IT is the one who achieves (or doesn't) compliance, and IT is the one who is held accountable. Strangely, we get held accountable by both sides. People actually give me grief as if I'm the one responsible for Sarbanes Oxley. It wasn't an IT guy that lied to everybody and wiped out their pension funds. And the really big secret of IT is that we feel the same way you do about the bureaucracy. We wish we could do away with it and get some things done.
I know that you're a smart guy; but I have my job for a reason, and it isn't because I can install BSD and set up a calendar. I could train a monkey for that. I have my job because I can also apply standards, evaluate a given system's impact in the environment, understand the policies and procedures (and the regulatory requirements from which they came), and keep it running when some obscure problem happens. You're certainly capable of doing all of those things; but you don't do them, because it's not your job, and you haven't got the knowledge and experience. I'm pretty smart myself, and I like to think I could be a good doctor; but I wouldn't set a leg just because I know how to mix plaster.
If you think the calendar is a priority, then walk over and beat on Sr. IT Mgt. They'll make it a priority and some IT tech will actually enjoy having a project he or she can complete that will make life easier for someone.