Chat is not e-mail. It's chat. It has a much different feel to it. One brings a different mental context to e-mail than they do to chat. It is accessed differently, and the use patterns are very different. Hence, solving "offline chat" by sending an e-mail... this is not a real solution.

You could argue that some of the characteristics we find in chat are not useful, counter productive, etc. And there is probably some truth to that. But that is a different argument than what you made.

Your "solution" does not work well with people on mobile. People on 2G/3G/LTE can alternate rapidly between online and offline. This kind of problem is easily solved by the chat server behaving like a store-and-forward device. Which is exactly what email is. But nobody likes to converse entirely using emails.

Chats have an important place because they allow us to say small things, quickly. What you suggest will result in no viable chat for mobiles, only emails. We can probably make a chat app that looks and feels like whatsapp but actually sends an email for every message. Dunno what problems this might have.

That's not really a protocol level solution. Sure, it works, if you know how to do it - but you don't automatically gain that knowledge "out of the box", you have to learn it. It is a real problem to anyone trying it out that doesn't understand why they can't send a message to someone who is offline.