Your Signal client has all the messages stored unencrypted (this is true because you can read them all, and search all the local messages).

What's preventing the client from dumping all the messages to a single file, encrypt it with a public key of the other running instance of Signal logged in to the same account, and send it, so that your other device can decrypt the file and import all the messages?

First people want full security, but when they encounter inconveniences that come with true full security, they start wanting convenience. What works best is a very very very good promise of security without e2ee, as shown by Telegram. It's users just know that Telegram is the most protected messages, and they are happy with it's advanced features.