There are already quite a few products that do exactly that. I work for one of those companies - we specialise in nurse scheduling (though also cover doctors and other healthcare staff), and have systems that both manage the hospital's demand (and using their own permanent and temp nurses), and systems for agencies to manage their 'inventory' to fill the gaps. You're right that it's a massive problem and therefore there is a good market here. The full on marketplace idea is also starting to be a possibility but isn't really a big deal yet.

It's also (unsurprisingly) really complex and I'm sure very far from their wine marketplace technology!