There are indeed a number of successful e-zines. In the fields of science fiction and fantasy -- where I do most of my writing -- two of the most prominent are Clarkesworld and Beneath Ceaseless Skies. They both use the same business model. The stories are published for free on the e-zine blog, then compiled into ebook format and sold on Amazon, Barnes & Noble, etc.
The thing to realize when starting up a magazine -- whether in print or in electronic format -- is that putting out regular issues is a lot of work, and quickly becomes a time drain. If the authors are paid, there's additional administrative hassle in terms of sending out contracts, payments, etc. And while it's definitely possible to earn significant revenue from an e-zine, in my experience most new start-ups tend to lose money for several years, at which point they either fold under or hit a fortuitous streak of publicity.