So let's say I wanted people on our network to be able to push video files to the Shield, for later viewing - is it possible? Would it even be possible to transfer through the device's USB port to an external hdd (making it somewhat like a NAS drive)?
Yes, apps like ES File Explorer can run an ftp server for easiest file transfer (you can set the root directory and permissions for just your videos folder). There are other similar solutions. If you're also running Kodi on it, the library can then be shared via UPnP for NAS-like functionality.
But to be honest a proper NAS with network shares that you can point something like Kodi to would be a better solution, IMHO.