How do I turn on the Steam Link without a Steam Controller?

It appears that yes, the only way to turn it on is to unplug it then plug it back in. There are a few threads on the Steam Community discussing this, here, here, and here (last one already mentioned by aman207).

It seems that, as described in the first thread, "the whole turning on and off thing is a bit of a mess at the moment". Speculated by many of the posters on the threads, perhaps there are bugs regarding turning on and off the system with those controllers, or maybe it is simply not implemented yet and will come with future updates.

As of several updates ago, turning on the Steam Link now works correctly with a wireless PS4 controller.