Eye-tracking animation plays correctly in Blender, but not in Unity

Hi there. I'm rigging my first character using the Rigify option in Blender, and have got a handle on weight painting and building an animation for export. So I've got an 'idle' animation for my character playing in Unity.
The only thing I was unhappy with were the eyes, which were being deformed as opposed to rotating. (They are separate, spherical objects in the Blender file). So I found this solution:
https://blenderartists.org/t/rigify-eyes-deform/1117896/5
..which suggests parenting the eye meshes to a by-default hidden bone at the eye position.
I did this and things in Blender look great. I imported into Unity again and the animation plays, but the eyes remained in position, while everything else in the character mesh moved around them. I then read that I should check the 'deform' box on the 'eye bones' each eye mesh is parented to in Blender. I did this and while the eyes start in the correct position now, they do not animate, remaining in place while the rest of the character moves around them. I assume it's something to do with them being parented to the MCH bones in the Blender file. Has anyone else encountered this problem? And, if so, what's the solution?

 

Sorry if this was more a Blender issue than a Unity one. I've figured out the problem and it was at the Blender end of things.

 

Hey! I found your thread in search of solutions to my problems trying to set up a smooth workflow using Rigify and Unity. Could you elaborate on your solution, in case it helps me (and hopefully other users?). I would believe many want to use Rigify with Unity, but there are so many hurdles, and so few solutions to be found