What is facebook's reason for not allowing this access? If I am a member of this group I have access to this information anyway via browser
That would be only you being able to see the posts then, fine. Two parties involved - you, and the other group members; everyone aware of who will be able to read the posts, all good.
But with an app, a third party gets involved - the app developer. And they could theoretically do anything with that data, that the app reads from the API on your behalf. (Sure, the app developer might be you in this special case here - but that is not the normal use case.)
And in a closed group people might discuss topics that are a little more sensitive than the current weather; think f.e. health problems or something like that. For sure not the stuff you’d want exposed to any third parties.
And therefor, the decision whether an app should be able to read the group’s feed posts, has been placed in the hands of the group admin(s).