Thanks, interesting.
No clue about this but wonder:
Whenever I see a pattern of "effect reduces, to close-ish to zero, if we start controlling for confounders - but stays nonzero", I wonder whether further related confounders that we simply cannot control for plausibly explain even the remaining nonzero. In this case: once you control "Socioeconomic status, parental education, and parental intelligence", and have still variation in breast-feeding left over, is that variation among those otherweise equal circumstances perfectly "random"? Probably not fully, and maybe the remaining confounders then also explain the rest of the gap.
The pattern might especially be explainable if the parents think there's a positive IQ effect, so in the best circumstances they'll try to breast feed, and even after controlling for the above effects, in less good circumstances they'll be less likely to breast feed. Maybe it's because they have less relaxed live, less health, or even slightly less mental energy or interest or care for the child. Bang: residual effect - but spurious.
Or is anything ruling that out?
Your hypothesis is plausible to me. The main factor pulling in the opposite direction is the prior that breastmilk is better because replacements don't fully emulate its bioactive compounds (e.g. antibodies) and nutritional profile (fats).
I haven't done the research here (and my vague impression from reading secondary sources is that it's similarly small-to-null), but the other thing to look at would be studies on breastfeeding and other outcomes (not intelligence-related). If we see a more of a positive correlation there, that's further evidence that the very small positive effects on brain development are real.
AFAIK the breast milk also contains antibodies which reduces the number of sick days in infants, on priors that should also have a (small) positive effect.
Socioeconomic status, parental education, and parental intelligence have strong effects on child IQ and are themselves correlated with breastfeeding practices. When studies ignore these confounders, they often find significant IQ increases in breastfed groups (5-8 points).
As more confounders are accounted for, the gap typically shrinks to 2-3 points. Some relevant studies:
However, the gold standard are sibling studies, where breastfed and non-breastfed siblings, who share the same parents and home environment, are compared. Here the effect shrinks further, though doesn’t go to zero.
The largest sibling-controlled study (Goldshtein et al.) unfortunately did not measure IQ. However, they did find that breastfeeding for at least 6 months is linked to about ~1-2 fewer children with developmental delays per 100, and ~7-13 fewer per 1000 with neurodevelopmental conditions. This was also confirmed in their sibling comparisons.