Mainly though, they limit the number of times a reader is pinged. I'm a big advocate for grouping all notifications, not just 'content' — only notify me if #_notifications > x. Issues can be a kind of inherent frequency limiter.
That said, they're absolutely not necessary and most online publishing (rightly) has no concept of issues.
"However, couldn't you just create artificial edges? For instance, showing the articles for the past 7 days?"
Yes — and I think more apps should allow for this.
Facebook is a great example (IMO, but clearly not in the opinion of many others) of doing just this — taking a near infinite stream of one-offs and grouping them based on your Facebook use frequency / habits. Twitter's (Summify) summary articles are another good implementation of this: what happened in my stream yesterday?