Nope, there are plenty of chats with e2ee and all the conveniences you’re mentioned.
Even Signal Desktop does not require the communication to pass through the phone (or even to have the phone around after setup). WhatsApp is the odd one in this respect.
Regarding Signal, they say this on their website:
> Signal Desktop is a Chrome app which links with your phone, so all incoming and outgoing messages are displayed consistently on all your devices.
https://signal.org/blog/signal-desktop/
That seems similar to what WhatsApp does, but I haven’t tried myself. Do you have more details on how that would work without passing by the mobile app?
The catch is that if your desktop is off, it won't be able to "catch up" later on any messages that it missed. Although I don't see why that's impossible to implement in principle.
Especially how they handle E2E encryption between multiple devices owned by the same user, without using one device as proxy.
Here’s how Wire works https://wire-docs.wire.com/download/Wire+Security+Whitepaper...
Here’s Signal’s multi device protocol https://signal.org/docs/specifications/sesame/
Here’s Viber’s security white paper https://www.viber.com/app/uploads/Viber-Encryption-Overview....