I see satipatthana as a near-synonym of jhana, as one is to develop it with vitakka-vicara, then with vicara, and then without either. So, samadhi without effort perhaps correlates with "...born of samadhi/unification of mind" and not "...born of seclusion" in the pericopes for the first two jhanas. Indeed, we can see the anapanasati lines "...calming bodily fabrication/calming mental fabrication/releasing the mind" as instructions for enacting this transition.
In any event, I'm disinclined to see a lack of will in jhana, as I consider that one directs ones mind to the destruction of the taints from within (fourth) jhana; the calming of fabrications is one thing, but the presence of right effort within the Samadhi category of the Path means that the suggestion that right effort drops away in right samadhi is wholly mysterious to me.
Finally, AN 9.37 is an odd one, in that it talks of the formless attainments without mentioning jhana at all. I'm fairly suspicious of conclusions about jhana built up on that sort of thing, but mileage varies.