Email is tough to kill because it's not a company or a product, it's a decentralized set of standards. Email sucks in a lot of ways, but it's simple, works everywhere, and it's baked into a lot of places (confirmation emails, password reset emails, alert emails, etc). There are better technologies out there like messaging and slack, but they're controlled by companies and not open/decentralized, and while they have integrations, they aren't baked into a lot of places (e.g. no Amazon confirmation slack messages or bank alerts through FB messenger).
Imagine trying to kill XML because JSON is better...nearly impossible. Email may slowly die over time, but I don't think it's likely you can kill it outright.
Yes, the fact that you can send email to people who aren't using your email system is the main thing that's keeping chat services stuck as part of a company's communication tools and not being the entire (or almost entire) thing.
I actually wrote the easy 50% of email back compatibility for a chat app once - it wasn't as hairy as I expected it was going to be. Everyone has a unique username (or at least username+org pair) so use that to generate an email address for them. Similarly, generate an email address for every group/room. When you receive an email for one of those addresses, use the from headers to find/create a new pseudo-user and dump the body of the email in as a chat message. When someone writes in chat that has a pseudo-user as one of the people watching that chat, send them an email with that text and throw in a few previous messages as a fake reply chain for context.
There are a lot of hairy details about thread management and making sure you don't send pseudo-users too many emails and dealing with all the messed up headers that different platforms send you and trying to only extract the actual message and not the signature/replies from an email and attachments so on that I only worked on a little before the whole thing was shuttered. But it was really cool. Email users tended not to notice that anything was different (they're used to email looking weird from various services) and chat users got to stay in chat that was a little awkward instead of having to switch to a different tool entirely to talk to people outside the organisation. (And, if all the different chat services started doing it, you'd get really janky chat federation for free!) And it lets you get a lot of stuff that gets made as Slack integrations for free, too. Why bother setting up a bot to post to chat when a Jenkins build fails when you can just give Jenkins your dev chat's email address and let it send emails?
Email is tough to kill because it's not a company or a product, it's a decentralized set of standards.
I think this is the answer to the question. Google and Microsoft e-mail monoculture could kill e-mail. Google Mail is well-known have incompatible IMAP extensions. I guess that if the two or three major players get a marketshare that is large enough, they'll start dropping more and more 3rd party mail ('just a glitch in the reputation system') until e-mail outside Google Mail and Exchange Online is not reliable enough anymore to bet your company business on.
Luckily, we are still far from that point, but with companies and universities switching to Google Apps and Office 365 en masse...
I guess the real question is "How feasible is it to remove or change the things that suck about email while keeping and possibly enhancing the good parts?" And I suppose the answer is that it's technically possible, but would be expensive, painful, and require a lot of cooperation.
Imagine trying to kill XML because JSON is better...nearly impossible. Email may slowly die over time, but I don't think it's likely you can kill it outright.