Email is a task manager by default; you're kind of forced into it by the fact that people will send you emails that are "actionable" in some way or another. Adding metadata about those emails into some other task-management system would just be redundant. Inbox takes the more sensible approach: presenting the actionable stuff from your inbox as tasks, so you can manage them. Certain emails are tasks either way; Inbox just lets you deal with those using a UX that fits their task-ness.

Personally, I think it would make a lot more sense if there was a clearer separation: emails only showing up in Inbox or Gmail, but not both. Any email that Inbox thinks is/was an actionable "task" should be tagged by Inbox's algorithms as such, and this should hide it from Gmail's default views. Then, you scrub your Inbox for what people want of you—and check your Gmail every once in a while to see what's piled up, but otherwise generally don't worry about it. Gmail would then just contain your non-actionable emails: the stuff where seeing it would never give you an anxiety attack. (It'd still let you do bulk operations to all your messages including the actionable ones, though.)

I think that "messages AI-sorted into separate bins, and then viewed through separate apps, but all in one email database" is actually quite a flexible architecture, and could be taken even further. For one example: email newsletters are another thing that's discrete from regular email†, and has different viewing needs and associated verbs. You could have a Google Reader-like interface that was really just plucking all the newsletters out of your email and displaying them cleverly.

But do note that Gmail would still be there, and still be actively maintained, even if you managed to create fancy abstracted email viewer apps covering the complete gamut of emails people receive. There are many "generic email" tasks that right now only Gmail performs, that Inbox has no plans to add support for: creating filters, adding forwarding accounts, etc. Effectively, Gmail is the "file manager" for your email database, the thing that deals with the unabstracted objects and lets you do maintenance tasks. You can't really throw it away any more than you can throw your file manager away.

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† You know why Google Reader was killed? Because RSS engagement and opt-in and open-rates are much lower than the same metrics concerning email newsletters. Anyone who wanted to make money publishing some regular piece of content cared about these metrics, and so ended up pushing their content as an email newsletter, leaving RSS to be solely the province of amateur personal bloggers. So, more and more, the people with power at Google who cared about consuming content just got it in their email, and generally forgot that RSS was a thing.

(There's a whole rant I could go on here about the fact that RSS was intended to be a supplier-side microformat from the start—a way to ease the burden of syndication to encourage custom blogging software to expose syndicated data—and there was never intended to be such things as "RSS Readers", but rather a much more active ecosystem of plumbing—like, for example, gateways to consume and transform feeds into things like emails. Such plumbing was intended to be employed by the website author, in the same way they'd employ e.g. a reverse-proxy like CloudFlare. RSS was meant to be the internal protocol between the blog and such infrastructure software: like FCGI/WSGI is between web servers and application servers. Picture a world before webhooks, where web servers are dumb and you can't expect to be able to "dial out" a server-side request from your app's crappy VPS; picture the architecture you'd use to hook e.g. Mailgun up to your website in this world. That's RSS.)