Let's make something clear: I love email. I have always enjoyed getting stuff in the (regular) mail, and when I first started sending and receiving emails on BBSes in the 90s, it was truly magical.
For a very long time I treated my inbox as a to do list. It's easy to see why: most mail apps are very similar to task management apps. There's a list of stuff. It can be labeled and organized and flagged and tagged. It has a bunch of things attached to it. But there's one important component missing for me: context.
As an example, I have an email in my inbox right now with the subject "Worksheet". It's something I forwarded to myself from my Gmail account, so it appears to be from... me. Given that, what do you think it is? Yeah, yeah, a worksheet. But about what? When is it due? Is it due, ever? Who is it truly from? What do I need to do? What's the first action I need to take on it?
If I don't get that out somewhere else, preferably in Things, then all of that is weighing down my cognitive load. I look at "From: Paul, Subject: Worksheet" and see about 8 different tasks I actually need to do. And in that moment it overwhelms me and feels like a block, a thing I need to take care of instead of something I may want to do.
In addition, the flow of what comes into and out of my inbox isn't controlled by me, not directly. I get junk mail (requires processing), links to stuff to read (goes to Instapaper), email newsletters (sits in the inbox... for now), and various messages. All of it requires some action, usually much more than "just" replying, and my email client says none of that. None.
Because of those hidden tasks, my inbox is a terrible to do list for me.
PS: about those tools
Obligatory: I could certainly find tools that aid in this. Back when I was at a company using Jira (shudder) I wrote a little AppleScript to process incoming bug emails and convert them to tasks in my Inbox in The Hit List (an abandoned to do list app). And yes, there are apps and plugins and things that will just go ahead and make your inbox a to do list. But for my workflow, I strongly prefer to treat my to do list as a separate item from the flow of email.