No where in the article did it state that the woman was taking drugs or had a food allergy. (In any case, allergens are a separate issue, unrelated to vaccination.)
1) Vaccines are injected into the arm muscle, not the blood.
2) The body is very efficient about destroying foreign proteins and nucleic acids (the vaccine components). In this case, the nucleic acid is messenger RNA which has, at most, a ten hour half life.
3) The immune reaction occurs at the site of injection, in the form of T-cells, B-cells, macrophages, etc., that come to the location in response to the presence of antigen.
4) In the case of vaccination, the challenge is to keep the antigen around long enough for the immune response to occur before the body destroys the antigen.
5) The milk of a vaccinated woman would contain antibody-producing B-cells, which protect the baby against disease. It not only is not harmful for the baby to consume the mother’s B-cells, the benefit that B-cells provide is one of the major reasons to advocate breast-feeding.
You are exhibiting tunnel vision.
When injecting into the muscle it is possible to get some of the dose into a small vein.
Only to someone who is focused on pushing a point of view thinks that it is irrelevant that many things can enter breast milk. It is relevant because it is wrong to be dogmatic ahead of targeted research in an area where there is incomplete understanding and a poor ability to predict.
Your statements are the kind that are given to the public, incomplete in the details and simplified to be used for general understanding. These kind of statements are like those given on food pyramids, which were constantly revised as our understanding evolved. The food pyramids were not only wrong but caused great harm. Humility is an important characteristic for a scientist as is caution in theorizing ahead of the data.