The study is suggesting that the marshmallow test is a test of mother without college degree, and not testing something about the children beyond that. [0]

And in general, my suspicion about psychology is, that since you can always invent an objection in the form of your (1) and (2), I doubt that statistic is an actually good tool to look at these tests. If we assume for a moment that we somehow get a ground truth of the experiment, by divine revelation or however, then I suspect that we would see that we need as many categories as there are children to get anything predictive.

[0] http://journals.sagepub.com/doi/abs/10.1177/0956797618761661