Signal Protocol already supports multi-device. We've encouraged them to enable that for Secret Conversations, and hopefully they'll continue to iterate towards support for e2e by default.
Although you could use local storage, you would have to do that with every browser you logged into and also somehow make it user-account-specific. I definitely see the challenges there for them.
What would you think about using a user's password to encrypt the browser keys and store them on Facebook's servers? Is that too large a compromise for usability?
What did the OP mean by "No secure way to verify code or store keys (in web browser) without routing through mobile" ?
This is why Signal and WhatsApp require the client to run on the phone - the phones are doing the decryption for the web apps.
This is flaky, consumes a lot of battery and generally is somewhat error-prone - probably not something FB wants to deal with.
Installing a malicious extension, tricking users into typing commands in the developer tools, XSSing FB, all of these are much easier to do than attacking a native app on a phone.
Certainly it can be important to know when you have a "no compromises" security option versus "mostly better than plaintext but maybe not secure".
It could be a UX judgment to not confuse users they have a "secure connection" when in fact they might not. Look at all the various attempts over the years browsers have made to keep the UX semi-reliable and easy for users to understand whether or not their SSL connection is secure.
You could even register only the browser, without having a smartphone at all.