I may have possibly stumbled on what I was looking for. A magazine that features new writing (as right now I find out about new writing sporadically here and there.)
You probably know about "Granta." Well, I forgot about it but I will go ahead and subscribe to it to see how it fares today (I think I used to read it about 7-8 years ago). But here is the news. Former "Granta" editor and NBCC president John Freeman launched his own thing, and the first edition of Freeman's: The Best New Writing on Arrival is out now. Of course it's not available at the library yet so I will ask them to buy it. It has short stories, new fiction, essays, poetry, and features both new writers and established behemoths. It comes out biannually, and in this issue there is a ghost story by David Mitchell, stories by Lydia Davis, Dave Eggers, Louise Erdrich, Haruki Murakami, new fiction by Laura van den Berg, Helen Simpson, Tahmima Anam, and essays by Aleksandar Hemon, Barry Lopez, and Garnette Cadogan.
Now, if there was a magazine like that for indie writers, I'd be happy. So far I haven't found one, but I will keep looking. Curating @ellowrites gives me a glimpse into new unpublished writing. Maybe it will grow into something bigger.