why would you take the risk of a stillbirth to avoid a slightly less painful birth?
Well, I don't think it's quite that simple.
First of all, induction vs waiting (forever) isn't the only choice. You could also have an elective c-section.
Second, all induction techniques are not the same as has been pointed out a lot already by other people on the thread. Some aren't as immediate as others so if the goal is to get the baby out as close to 40 weeks as possible, you might choose something different compared to if you just want to jog things along a bit.
Third, dates can be incorrect and there is a natural tendency for spontaneous labour right up to 42 weeks, isn't there? In fact the average time for a first time mum to go into labour is naturally a little bit later - something like 40+6 I think. So it might well be premature to induce labour by or at 40 weeks for some women or even most first time mums. There is also monitoring you can do first of all to check how favourable a woman's body is likely to be towards the induction, and second to check on the health of the baby and the placenta. I don't know offhand how effective this is (actually I think this is what that review is looking at with "expectant management" - so perhaps not that effective).
Fourth, none of this is happening in a vacuum. Some of the stillbirths happening after 40 weeks might well be caused by the induction, as induction before 40 weeks is very rare so most pre-40 week labours are going to be spontaneous whereas more 41+ or 42+ week labours are going to be induced ones.
Fifth, an induced labour is riskier than a spontaneous one, which needs to be weighed up against the increased risk of stillbirth - this is as I understand it why the current recommendation is to wait 10-14 days after the due date before scheduling an induction, rather than inducing immediately at 40 weeks. There are also other risks such as injury and trauma to the mother, which aren't insignificant. It can be easy to say well I'd take any injury or trauma if it means my baby will live - and that might well be true, but if for example there is a 10x increased risk of injury vs a 0.2x increased risk of stillbirth this might not be a reasonable comparison to make. Or it might be. That's why the individual patient needs to have a conversation about risks with her midwife or consultant in the case where induction is being suggested.
Sixth, we're not talking "slightly more painful" - but quite significantly more painful (going by anecdotal evidence, anyway) with an increased risk of lasting injury (documented) or trauma (anecdotal? Possibly studied, I don't know). But in any case as earlier stated, it's not a choice between induction or waiting, but actually between induction, elective section or waiting. Elective section is not painful (apart from the recovery perhaps) and tends not to be traumatic, particularly compared to instrumental delivery or emergency c-section, both of which are higher likelihood with induction. Of course it's major surgery, so does involve recovery and does probably count as an injury in terms of scarring and potential damage, but it's usually something people recover from completely as opposed to some of the more catastrophic bladder, bowel or vaginal injuries which can be caused by instrumental delivery.