Why tie it to a phone at all? Why not have this Chrome app work standalone? Apple's messaging app uses end-to-end encryption and it can be routed to multiple devices. I would expect Signal to strive to attain parity with Apple's offerings.

Also, the naming of this app is pretty disingenuous. This app should be called "Signal Remote", since it's not running the actual Signal comms on my desktop, it's remotely controlling Signal on my phone.