My peers and I work on a language centered around "constructive data modeling" (first time I hear it called that). We implement integers, and indeed, things like non empty lists using algebraic data types, for example. You can both have a theory of values that doesn't rely on trapdoors like "int32" or "string", as well as encode invariants, as this article covers.
As I understand it, the primary purpose of newtypes is actually just to work around typeclass issues like in the examples mentioned at the end of the article. They are specifically designed to be zero cost, because you want to not pay when you work around the type class instance already being taken for the type you want to make an instance for. When you make an abstract data type by not exporting the data constructors, that can be done with or without newtype.
As I understand it, the primary purpose of newtypes is actually just to work around typeclass issues like in the examples mentioned at the end of the article. They are specifically designed to be zero cost, because you want to not pay when you work around the type class instance already being taken for the type you want to make an instance for. When you make an abstract data type by not exporting the data constructors, that can be done with or without newtype.