They probably wouldn't die for their kids. The most common situation where this gets tested is famine, and when famine is bad enough for long enough, parents collectively decide it's OK to eat kids to keep functioning adults in the society for a chance at group survival. See the Khmer rouge famines for example.

Now I'm just guessing this part, but I feel like maybe the people who would actually die for their kids are the same people who are trying their best to eat healthy, apply critical thinking to what they watch on TV, etc.