Is the issue also a lack of available sensor data?

Our own nerves produce a huge amount of data. Our eyes and balance also contribute to a very rich picture to enable movement and kinematics. This is all fused into usable feedback.

Arms on fixed axis with disconnected vision feel like we are still very far away from true robot training.

That is part of the problem. Yes, humans are absolutely studded with sensors, every hair you've got, every nerve ending. It's amazing that all that information gets processed as fast as it does even though the 'base clock' is rather low. There is a ton of pre-processing done on those signals though (for instance: the optical nerve is pretty much an extension of the brain, how is that for a high speed, high bandwidth interface). Then there are the reflexes, semi-autonomous parts of the nervous system that can react faster than the brain to deal with safety related concerns. For a very interesting biology angle have a look at octopi.

Exactly. I suspect the problem isn't just the amount of training data, but how much of the relevant information is missing from it.