There was one missing link in the milk hypothesis. 

It was necessary to also show that breast feeding was declining at the time. This was, indeed, the case with the onset of the industrial revolution and women becoming earning members of the family. But it also appears to have been an "upper class" preference.

The measles mortality rates would likely not have been so acute had mothers not given up on breastfeeding to the extent they did. 

Btw, I've read somewhere that Women Health Visitors used to encourage mothers to revert to breast feeding as far as possible.