In the podcast you mention breastfeeding quickly but really don't spend any time on this subject. In fact there are many studies which investigate the link between breastfeeding and food allergies. Like other studies on food allergies it is limited based on the ability to control the study but there seems to be an obvious trend among the various studies which does back up some of the findings. The American Academy of Pediatrics recommends breast feeding for at least the first month. I haven't had time to read all the studies but I definitely would consider this among the possible factors when it comes to food allergies.

As to why some schools ban peanuts and other foods, this is just side effect of a very small group of parents with loud voices. Children with food allergies should just carry an epi and not expect everyone around them to cater to their health issues.