Nothing unusual about the CDC noting that breastfeeding can fight infection in infants. It's been known for a long time, which is why many well-informed sources recommend breastfeeding.
Where people go wrong is in assuming that breastfeeding is a _substitute_ for vaccination. A lot of the important maternal antibodies are transferred to the infant in utero (in the womb), not through breast milk. These antibodies provide short-lived immunity (often just a few months worth - rather than the long-term protection vaccines provide). Add to that the variable level of immunity mothers have to vaccine-preventable diseases, so protection through breastfeeding is uncertain.
All this means that breastfeeding is a good _complement_ to immunization to keep infants and kids healthy.