The cost for mech turk:
We paid something like $10 per hour and people loved our tasks. We paid a bit more to make sure our tasks were completed well. The main thing was just making the data collection app as efficient as possible. If you pay twice as much but collect 4x the data in the task, you doubled your efficiency.
Yeah I think its impossible to get good gaze accuracy without observing the device reflection in the eyes. And you will never, ever be able to deal well with edge cases like lighting, hair, glasses, asymmetrical faces, etc. There's just a fundamental information limit you can't overcome. Maybe you could get within 6 inches of accuracy? But mostly it would be learning face pose I assume. Trying to do gaze tracking with a webcam of someone 4 feet away and half offscreen just seems Sisyphean.
Is EyesOff really an important application? I'm not sure many people would want to drain their battery running it. Just a rhetorical question, I don't know.
With the baby autism screener its difficult part is the regulatory aspect. I might have some contacts in Mayo Clinic that would be interested in productizing something like this though, and could be asked about it.
If I were you, I would look at how to take a mobile photo of an iris, and artificially add the reflection of a phone screen to create a synthetic dataset (it won't look like a neat rectangle, more like a blurry fragment of one). Then train a CNN to predict the corners of the added reflection. And after that is solved, try the gaze tracking problem as an algebraic exercise. Like, think of the irises as 2 spherical mirrors. Assume their physical size. If you can locate the reflection of an object of known size in them, you should be able to work out the spatial relationships to figure out where the object being reflected is relative to the mirrors. This is hard, but is 10-100x easier than trying end-to-end gaze tracking with a single model. Also nobody in the world knows to do this, AFAIK.