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Earlier quoted context omitted.
That only applies to people who have iOS devices, so it still leaves a whole swatch of people out of the equation.
Doe you not mean swath or swathe rather than swatch? (Genuinely curious I think this might be a locality thing)
Text/IM really doesn't belong in the inbox in my opinion, because as others have stated these are in a different category of communication, with different urgency levels. IM and SMS are for when you need to get in touch instantly or near-instantly, email is for more asynchronous and/or one-way communication.
That said, IFTTT sounds like the perfect solution: https://ifttt.com
"If This Then That", or IFTTT, lets you trigger actions from events. So, if you get a Twitter mention, or an SMS, you could have it email it to you.
Mozilla Raindrop, now inactive used to do that. https://mozillalabs.com/en-US/raindrop/
Know if the open-sourced it or is it just dead and gone?
Do you need a unified inbox where you can read all the messages, or do you actually want to be able to search across all your inboxes (like you could with Greplin) or aggregate notifications that you received a new message (Chime)?