Anyway - I too have an overflowing inbox. I used to aim for Inbox zero but slipped and am living with it.
One discipline I do maintain though is that the Inbox should always be actionable. It's my todo list.
So - how do I handle non-actionable, non-urgent stuff?
1. If it will be urgent and actionable I snooze it. 2. If it's of interest but not urgent and has a web representation then I open a browser tab for later.
Newsletters nearly always fall into category 2. Unless I can just skim them for interesting links and archive them.
And a lot of the time I unsubscribe - as soon as I realise I don't really want to read it. I get most of my news from HN or various subreddits.
I think there's about 3 or 4 regular newsletters I tolerate. And I'm more tolerant of infrequent emails (new features for products I'm interested in etc)