I fail to understand how the evidence is convincing that the genes followed diligently such a commonsense assumption that after weaning, the enzyme would be no longer needed and then made unavailable.
Humans consume many things that are not perfectly digestible without any great ill effect.
The domestication of mammal herds makes available a source of excess milk but by itself doesn't prove that the enzyme had remained switched off thousands of years before.
No it doesn't. That's why the researchers mentioned in the article conducted the genetic experiment they did, in order to date the mutation.