Great Britain may have been named after one of the isles, but the controversy arises because the UK spans (part of, but not all of) the two main isles.
> Has anyone proposed a viable alternative?
I don't know what would count as "viable" if you reject "Britain and Ireland" and "Atlantic Archipelago".
> Great Britain may have been named after one of the isles
Great Britain is the largest island of the isles. The UK's full name is The United Kingdom of Great Britain and Northern Ireland; ie there's no political entity of Great Britain (and never has been).
Indeed. I suppose if Ireland's full name were "The Republic of Little Britain" there wouldn't be a controversy about the term "British Isles".
> there's no political entity of Great Britain (and never has been).
Unless you include the Kingdom of Great Britain (1707–1801):
Has anyone proposed a viable alternative?