"W zimie" also works.
I'd say that it's not exactly Instrumental, it's an adverb that looks like the Instrumental version of the noun. Just like "wieczorem". And yes, I think I'd be more likely to say "Zimą" than "W zimie" in such a sentence.
The acceptable versions, according to this website (http://ulalupinska.pl/miniporadnia-zima-lato/) are: wiosną/na wiosnę, latem/w lecie, jesienią/na jesieni and zimą/w zimie.
I wouldn't think about "na" as "at". In fact for a moment I was really confused, thinking that you're asking about English - why would you? :D
I don't know the reason. The website I linked also says that the author couldn't find any explanation 'why'. It is how it is, I'm afraid...