The most important meal of the day

There is some fascinating research reported in today's Western Mail. It says the free primary school breakfast scheme might have nutritional benefits, but does not improve educational attainment.
Ministers have repeatedly said that children work harder, behave better and concentrate for longer with a decent breakfast inside them. This is undoubtedly true.
So why does the scheme not affect education? Another finding suggests those same children were eating breakfast at home anyway. The study found "areas where pupils had previously had breakfast at home were more likely to take-up the free breakfast scheme".
I wonder if it is the case that the only kind of parent who will pressurise a school and local council into participating in the free breakfast scheme is the kind of parent who will make sure his or her child eats breakfast anyway.
Primary school breakfast clubs are great. Lots of schools have told me so. But am I wrong to suspect that a child who does not get fed in the morning at home is unlikely to have someone who will write to the school governors on their behalf asking for free breakfasts to be provided?