Not very likely any time soon. There was a long debate about it and they decided that putting your trust entirely in the CA system and Signal's servers every time to not serve a malicious client that couldn't be validated by the user wasn't acceptable.

Why can't I self host a web app.

I just don't want any more applications open.

I'm CA's will be bypassed soon. There's a lot of brain attention on this.

It has a desktop app for MacOS, Linux and Windows. https://github.com/signalapp/Signal-Desktop

It's as "native" and "desktop" as... well... ekhm: "Signal Desktop is an Electron application that links with your Signal Android or Signal iOS app."