it's done today, kinda

Implanted chips are too small to send signals readable from any big distance, so like pets the child would have to be brought to a facility that has equipment that can read the chip. And what kidnapper would bring the child they kidnapped to such a facility? It might be useful in the case of John Doe runaways, or lost children with no ID who won't talk. But those are rare scenarios. What'd be much more useful is a larger transmitter the child has in their pocket that can be tracked as needed (if they don't come home from school, or whatever.) Cellphone software already exists that can be used for that purpose, although it depends on the child not ditching the cell phone.

Unless it were mandatory, only higher-class children would get chipped. And I'm pretty sure it's the lower-class children who are more likely to be kidnapped (by their parents.) If it were mandatory there would quickly be goal displacement at work, using them for tracking for other purposes. I could imagine it getting pushed the same way it's pushed for children to have regular photos taken, in case they're kidnapped/missing. Or they'll push doctors to push it on parents, combining it with other infant vaccinations.