Not everyone would agree that “I want it to work” is a correct restatement of deontic modality. (The one I use when wanting to avoid the ambiguity of “should” is “it had better work”.)

Yes, that is a better way of phrasing it. Changed.

That effect is due to the fact that you're forced to use your System 2. It probably disappears after you become too fluent in the language (for example, FWIW, I don't ‘feel’ that happening with English).

Unfortunately, this seems likely.