My daughter is lactose intolerant (diagnosed at 11 months), but we were advised to allow some access so that she would not become allergic. We found that by removing milk, cream, sausages, and cows milk cheese, we were OK, but the odd small ice cream or slice of pizza would be OK if it was only very occasionally. We used to get ewes milk and cheese.
My cousin was obsessively scrupulous with her son and he developed allergic reaction. It does seem to be modern thinking with nut allergies that a very low exposure is a good idea, but not sure I would want to risk it, would you?
Coeliac is also a failure to digest, so an intolerance not an allergy, but I might think that total abstinence might also develop an allergy?