I can see this theory working in several scenarios, despite (or perhaps rather because of) the relative fuzziness of its description (which is of course the norm in psychological theories so far). However I have personal experiences that at least at face value don't seem to be able to be explained by this theory:

During my breaks I would read textbooks, mostly mathematics and logic, but also branching into biology/neuroscience, etc. I would begin with pleasure, but if I read the same book for too long (several days) my reading speed slows down and I start flipping a couple pages to see how far it is till the next section/chapter. So to me it this seems not like a motivation shift from "have-to" to "want-to", but rather the brain's getting fatigued at parsing text/building its knowledge database, and subjectively I still want to keep reading, and advancing page by page still brings me pleasure, but there's something "biological" that keeps me back (of course everything about me is biological, but I mean it in a metaphorical way, that it feels quite distinct from the motivational system that makes me want to read).

Now I have found an easy way to snap out of it: simply switch the book/subject. Switching from math to biology/neuroscience works better than switching from math to math (e.g. algebra to topology, category theory to recursion theory, etc), but the latter can still recover some of the mental resistance built up. I don't see how this can fit in the framework of "have-to" and "want-to". Nobody's forcing me to read these books; it's purely my desire. If the majority of executive function can be explained in such a way as expounded by the paper, then I do not see how switching subject of reading can make such a big difference.

Of course I may be an outlier here, or I'm misunderstanding what constitutes "willpower" or not. Feel free to offer your opinions.

Either way, I'm glad that this is an active area of research. I'm quite interested in motivation myself.