> writing a fresh map function for each data type

You have to do that in Haskell as well (where »you« can be you or the compiler via DeriveFunctor); the key difference is at the use site: there is one map function (called fmap) that does the mapping, whereas in Elm (and pretty much all other languages that don’t have higher kinds while we’re at it) you have to use the specific map function for that type.