They aren't conflicting goals. Multi-device E2EE is available in protocols like Matrix and WhatsApp.
In the simplest case multi-device E2EE can be implemented as a group chat, and when you add a new device to your account you automatically add it to all of your rooms. So any protocol that supports mutli-user E2EE can support multi-device E2EE. Of course there are more efficient implementations.
it would involve you exporting the key, storing it somewhere, and then importing it to a new device from where you stored it.
Yes, you need to have a copy of the key, if the last copy is lost any E2EE solution will fail closed. If you have multiple devices this is probably already solved. (For example Matrix where when you log in with a new device it will ask you to verify from an existing device.)
But the point stands that if I am on vacation with a laptop and a phone and I lose my phone with proper multi-device I can continue to use my laptop seamlessly. (It already has a key)
You can also make "offline" backups and import to new devices. This may be less convenient but it can be easier to make offline backups than having globally distributed full computers. There are other solutions as well like escrow where a key is protected by a password or HSM devices. Although these are not as strong as never giving the key to a third-party.