Gödel's theorem for Peano Arithmetic shows that (under consistency hypothesis on PA) there is a statement which cannot be proved or disproved within PA that is true under the standard model (naturals). An immediate consequence of Gödel's theorem is that there must be some different (nonstandard) model(s) of PA. Gödel's method to accomplish incompleteness of PA involves a noticeable amount of machinery.
I know that there is surely a statement in FO group theory (the statement that in a rough interpretation would intuitively say that the group is abelian) which is easy to see that is not provable nor disprovable because it is true in some models and false in other models. Am I missing something, or this second approach is more natural and effortless than the whole Gödel's construction with respect to PA?