I don't think food scarcity/availability/price plays any appreciable role in developed nations birth rates but I'd be willing to consider evidence that it does.
I think price is a factor. It isn’t uncommon for someone to say, “another mouth to feed.”
When adding people to a family, food is probably the biggest non-negotiable cost going up.
I’d also expect to see a strong correlation between family size and the percent of meals eaten at home. Going out to eat quickly becomes unaffordable as family size grows.