The protocol of provisioning shared key to multiple devices?
This looks likely a classic server-multiple client scenarios, I had the similar situation:
Depend on real application, some of my thought:


For most case, IoT devices does not communicate to server at the same time? so server should be able to setup secure key for each of IoT devices through ECDH.
To have a common key for all devices looks too dangerous, that means as soon as one IoT device is compromised, hacker will be able to hijack all IoT devices..
It is only safe for each IoT device to use a common AES key--when you have physical access to them--for exp, in factory, in Lab, before shipment .. after it is been deployed to field, you must consider the different key for each device. Someone also burn a HW AES key into IoT device ROM, but still that have to be different for each of the IoT device, so one compromise does not jeopardize all devices.
Given ECDH could be too heavy for a IoT devices (usually take few back-forth communication), my suggestion would be to consider once time ECDH when IoT device is been activated, and using Ephemeral ECDH (ECDHE) which does not request regenerated ECC keys.