Not sure where the "mothers are secretly killing their babies and the doctors are covering for them" came from but it only seems to be repeated by those without kids.
Probably from vagary of the syndrome name/description. For me, if SIDS is accidental suffocation, then I'd expect the autopsy to state that they died of accidental suffocation, not that-- effectively-- they "died of sudden death".
...but that's actually what happens. SIDS is a "default" diagnosis-- it's not what's going on the cert if a specific causal chain can be discerned. It's literally "we have no idea how this happened" given a fancy name.
Add on to the fact that postpartum depression craziness is a thing, there's very little in the way of conclusions for this syndrome (this is the big part; myself, I cannot disagree with the suspicion about a long known issue like this not being investigated to the bone with more results than some conjecture about sleeping posture and brain deformities), and-- fortunately-- not too many infants at all are posthumously diagnosed with SIDS, and I can at least see where the theory might have came from. Adding fuel to all these components is that the attitudes around this are seen as feminist in nature-- as in, "SIDS" is seen as a way to dismiss the idea that a woman could actually kill her own child
never mind the five Holocausts worth of children killed since 1973 in the U.S. alone because they want to elevate women as virtuous figures. That said, from there, given that we still get reports of women killing their infant children or throwing them away to die of exposure, I reckon one may lead themselves to the belief that either the zeitgeist that fuels this inclination isn't spread contiguously enough or it's unwilling to jump to the defense of "too obvious" cases.
Being critical, though, there's a fair bit of shifting around on culpability in this theory, now that I think about it. From the same proponents, the proposed motive can drift from "the mother is smothering the baby" to "the mother is in a sleep-deprived daze and forgot to lay down her child the right way" or "the mother accidentally smothered the child in a sleep-deprived or post-partum depressive daze after realizing that the child 'quiets down' if she holds him a certain way". Maybe that's not a shift as much as it's a clarification and I misunderstood the original argument-- perhaps the actual idea is that women are
manslaughtering their children and the doctors are covering up for them rather than that they're
murdering them and the doctors are covering up for them. In which case, the indignation presumably comes from the aforementioned conjecture of feminist motivations.